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A364343 Expansion of Sum_{k>0} k * x^k/(1 + x^k)^3. +0
0
1, -1, 9, -12, 20, -12, 35, -60, 72, -30, 77, -132, 104, -56, 210, -256, 170, -117, 209, -320, 378, -132, 299, -672, 425, -182, 594, -588, 464, -360, 527, -1040, 858, -306, 910, -1224, 740, -380, 1170, -1640, 902, -672, 989, -1364, 1890, -552, 1175, -2928, 1470, -775, 1938, -1872, 1484, -1080, 2090 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = (n/2) * Sum_{d|n} (-1)^(d+1) * (d+1) = (n/2) * (A002129(n) + A048272(n)).
MATHEMATICA
a[n_] := DivisorSum[n, (-1)^(# + 1)*(# + 1) &] * n/2; Array[a, 55] (* Amiram Eldar, Jul 20 2023 *)
PROG
(PARI) my(N=60, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1+x^k)^3))
CROSSREFS
Cf. A320900, A364351.
Cf. A002129, A048272, A309731.
KEYWORD
sign,new
AUTHOR
Seiichi Manyama, Jul 19 2023
STATUS
approved
A364351 Expansion of Sum_{k>0} k^2 * x^k/(1 + x^k)^3. +0
0
1, 1, 15, -6, 40, 12, 77, -60, 180, 30, 187, -120, 260, 56, 630, -376, 442, 117, 551, -340, 1218, 132, 805, -1104, 1325, 182, 1998, -672, 1276, 360, 1457, -2032, 2970, 306, 3290, -1710, 2072, 380, 4134, -3080, 2542, 672, 2795, -1672, 7830, 552, 3337, -6816, 4998, 775, 7038, -2340, 4240, 1080 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = (n/2) * Sum_{d|n} (-1)^(n/d+1) * (d+n) = (n/2) * (A000593(n) + n * A048272(n)).
MATHEMATICA
a[n_] := DivisorSum[n, (-1)^(n/#+1) * (#+n) &] * n/2; Array[a, 55] (* Amiram Eldar, Jul 20 2023 *)
PROG
(PARI) my(N=60, x='x+O('x^N)); Vec(sum(k=1, N, k^2*x^k/(1+x^k)^3))
CROSSREFS
Cf. A320900, A364343.
Cf. A000593, A048272, A309732.
KEYWORD
sign,new
AUTHOR
Seiichi Manyama, Jul 19 2023
STATUS
approved
A364298 Square array read by ascending antidiagonals: T(n,k) = [x^k] 1/(1 + x) * Legendre_P(k, (1 - x)/(1 + x))^(-n) for n >= 1, k >= 0. +0
0
1, 1, 1, 1, 3, 19, 1, 5, 73, 721, 1, 7, 163, 3747, 49251, 1, 9, 289, 10805, 329001, 5370751, 1, 11, 451, 23623, 1179251, 44127003, 859748023, 1, 13, 649, 43929, 3100001, 190464755, 8405999785, 190320431953, 1, 15, 883, 73451, 6751251, 589050007, 42601840975, 2160445363107 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
In the square array A364113, the k-th entry in row n is defined as [x^k] 1/(1 - x) * Legendre_P(k, (1 + x)/(1 - x))^n. Here we essentially extend A364113 to negative values of n.
The two types of Apéry numbers A005258 and A005259 are related to the Legendre polynomials by A005258(k) = [x^k] 1/(1 - x) * Legendre_P(k, (1 + x)/(1 - x)) and A005259(k) = [x^k] 1/(1 - x) * Legendre_P(k, (1 + x)/(1 - x))^2 and thus form rows 1 and 2 of the array A364113
Both types of Apéry numbers satisfy the supercongruences
1) u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r))
and the shifted supercongruences
2) u(n*p^r - 1) == u(n*p^(r-1) - 1) (mod p^(3*r))
for all primes p >= 5 and positive integers n and r.
We conjecture that each row sequence of the present table satisfies the same pair of supercongruences.
LINKS
Table of n, a(n) for n=1..44.
EXAMPLE
Square array begins
n\k| 0 1 2 3 4 5 6
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 | 1 1 19 721 49251 5370751 859748023
2 | 1 3 73 3747 329001 44127003 8405999785
3 | 1 5 163 10805 1179251 190464755 42601840975
4 | 1 7 289 23623 3100001 589050007 152184210193
5 | 1 9 451 43929 6751251 1479318759 434790348679
6 | 1 11 649 73451 12953001 3219777011 1062573281785
MAPLE
T(n, k) := coeff(series(1/(1+x)* LegendreP(k, (1-x)/(1+x))^(-n), x, 11), x, k):
# display as a square array
seq(print(seq(T(n, k), k = 0..10)), n = 1..10);
# display as a sequence
seq(seq(T(n-k, k), k = 0..n-1), n = 1..10);
CROSSREFS
A364299 (row 1), A364300 (row 2), A364301 (main diagonal), A364302 (first subdiagonal). Cf. A005258, A005259, A143007, A364113.
KEYWORD
nonn,tabl,easy,new
AUTHOR
Peter Bala, Jul 18 2023
STATUS
approved
A364302 a(n) = [x^n] 1/(1 + x) * Legendre_P(n, (1 - x)/(1 + x))^(-n-1) for n >= 0. +0
0
1, 3, 163, 23623, 6751251, 3219777011, 2313306332191, 2337707082109071, 3163417897474821763, 5524913023443862515019, 12101947272421487464092429, 32493996621780038121738419591, 104964758754905547830609842389527, 401618040258524641485654323795309235 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
First subdiagonal of A364298.
LINKS
Table of n, a(n) for n=0..13.
FORMULA
Conjectures:
1) the supercongruences a(p) == 2*p + 1 (mod p^3) hold for all primes p >= 5 (checked up to p = 101).
2) the supercongruences a(p - 1) == 1 (mod p^4) hold for all primes p >= 3 (checked up to p = 101).
3) more generally, the supercongruences a(p^k - 1) == 1 (mod p^(3+k)) may hold for all primes p >= 3 and all k >= 1.
MAPLE
a(n) := coeff(series( 1/(1 + x) * LegendreP(n, (1 - x)/(1 + x))^(-n-1), x, 21), x, n):
seq(a(n), n = 0..20);
CROSSREFS
Cf. A364117, A364298, A364301.
KEYWORD
nonn,easy,new
AUTHOR
Peter Bala, Jul 18 2023
STATUS
approved
A364301 a(n) = [x^n] 1/(1 + x) * Legendre_P(n, (1 - x)/(1 + x))^(-n) for n >= 0. +0
0
1, 1, 73, 10805, 3100001, 1479318759, 1062573281785, 1073267499046525, 1451614640844881665, 2534009926232394596267, 5548110762587726241026801, 14890865228866506199602545427, 48084585660733078332263158771313, 183923731031112887024255817209295155, 822427361894711201025101782425695273529 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Main diagonal of A364298 (with extra initial term 1). Compare with A364116.
Compare with the two types of Apéry numbers A005258 and A005259, which are related to the Legendre polynomials by A005258(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x)) and A005259(n) = [x^k] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x))^2.
A005258 is the main diagonal of A108625 and A005259 is the main diagonal of A143007.
LINKS
Table of n, a(n) for n=0..14.
FORMULA
Conjectures:
1) a(p) == 2*p - 1 (mod p^4) for all primes p >= 5 (checked up to p = 101).
More generally, the supercongruence a(p^k) == 2*p^k - 1 (mod p^(3+k)) may hold for all primes p >= 5 and all k >= 1.
2) a(p-1) == 1 (mod p^3) for all primes p except p = 3 (checked up to p = 101).
More generally, the supercongruence a(p^k - p^(k-1)) == 1 (mod p^(2+k)) may hold for all primes p >= 5 and all k >= 1.
MAPLE
a(n) := coeff(series( 1/(1 + x) * LegendreP(n, (1 - x)/(1 + x))^(-n), x, 21), x, n):
seq(a(n), n = 0..20);
CROSSREFS
Cf. A005258, A005259, A108625, A143007, A364116, A364298, A364302.
KEYWORD
nonn,easy,new
AUTHOR
Peter Bala, Jul 18 2023
STATUS
approved
A364300 a(n) = [x^n] 1/(1 + x) * Legendre_P(n, (1 - x)/(1 + x))^(-2) for n >= 0. +0
0
1, 3, 73, 3747, 329001, 44127003, 8405999785, 2160445363107, 720972846685225, 303256387595475003, 157007652309393485073, 98141188253799911132091, 72882030213423405890701449, 63436168183711463443127520699, 63968150042375034921379294100073, 73985402858435691329113991048739747 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Row 2 of A364298.
Compare with the Apéry numbers A005259, which are related to the Legendre polynomials by A005259(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x))^2.
A005259 satisfies the supercongruences
1) u (n*p^r) == u(n*p^(r-1)) (mod p^(3*r))
and the shifted supercongruences
2) u (n*p^r - 1) == u(n*p^(r-1) - 1) (mod p^(3*r))
for all primes p >= 5 and positive integers n and r.
We conjecture that the present sequence also satisfies the supercongruences 1) and 2).
LINKS
Table of n, a(n) for n=0..15.
FORMULA
Conjectures:
1) 17*a(p) - 11*a(p-1) == 40 (mod p^5) for all primes p >= 7 (checked up to p = 101).
2) for r >= 2, 17*a(p^r) - 11*a(p^r - 1) == 17*a(p^(r-1)) - 11*a(p^(r-1) - 1) (mod p^(3*r+3)) for all primes p >= 5.
3) a(p)^(3*17) == a(1)^(3*17) * a(p-1)^11 (mod p^5) for all primes p except p = 5 (checked up to p = 101).
4) for r >= 2, a(p^r)^(3*17) * a(p^(r-1) - 1)^11 == a(p^(r-1))^(3*17) * a(p^r - 1)^11 (mod p^(3*r+3)) for all primes p >= 5.
MAPLE
a(n) := coeff(series( 1/(1 + x) * LegendreP(n, (1 - x)/(1 + x))^(-2), x, 21), x, n):
seq(a(n), n = 0..20);
CROSSREFS
Cf. A005259, A364298, A364299, A364301, A364302.
KEYWORD
nonn,easy,new
AUTHOR
Peter Bala, Jul 18 2023
STATUS
approved
A364299 a(n) = [x^n] 1/(1 + x) * Legendre_P(n, (1 - x)/(1 + x))^(-1) for n >= 0. +0
0
1, 1, 19, 721, 49251, 5370751, 859748023, 190320431953, 55743765411043, 20884452115700251, 9745388924112505269, 5543574376457462884111, 3776677001062829977964007, 3036161801705682492174749691, 2844274879825369072829081331519 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Row 1 of A364298.
Compare with the Apéry numbers A005258, which are related to the Legendre polynomials by A005258(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x)).
A005258 satisfies the supercongruences
1) u (n*p^r) == u(n*p^(r-1)) (mod p^(3*r))
and the shifted supercongruences
2) u (n*p^r - 1) == u(n*p^(r-1) - 1) (mod p^(3*r))
for all primes p >= 5 and positive integers n and r.
We conjecture that the present sequence also satisfies the supercongruences 1) and 2).
LINKS
Table of n, a(n) for n=0..14.
FORMULA
Conjectures:
1) 13*a(p) - 7*a(p-1) == 6 (mod p^5) for all primes p >= 3 (checked up to p = 101).
2) for r >= 2, 13*a(p^r) - 7*a(p^r - 1) == 13*a(p^(r-1)) - 7*a(p^(r-1) - 1) (mod p^(3*r+3)) for all primes p >= 5.
3) a(p)^13 == a(p-1)^7 (mod p^5) for all primes p >= 3 (checked up to p = 101).
4) for r >= 2, a(p^r)^13 * a(p^(r-1) - 1)^7 == a(p^(r-1))^13 * a(p^r - 1)^7 (mod p^(3*r+3)) for all primes p >= 5.
MAPLE
a(n) := coeff(series( 1/(1 + x) * LegendreP(n, (1 - x)/(1 + x))^(-1), x, 21), x, n):
seq(a(n), n = 0..20);
CROSSREFS
Cf. A005258, A364298, A364300, A364301, A364302.
KEYWORD
nonn,easy,new
AUTHOR
Peter Bala, Jul 18 2023
STATUS
approved
A364244 a(n) = A143007(2*n-1, n-1) for n >= 1. +0
0
1, 25, 1441, 107353, 9073501, 826861993, 79219824685, 7865844936025, 802198564524325, 83532710607121525, 8844234718023010681, 949244022625120188265, 103044177225432902852641, 11293765432962617876667253, 1248038875078327818254657941 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The sequence of Apéry numbers A005259 forms the main diagonal of A143007, i.e., A005259(n) = A143007(n, n). The Apéry numbers satisfy the supercongruences A005259(n*p^r) == A005259(n^p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers n and r. We conjecture that the present sequence satisfies the same supercongruences.
More generally, for positive integers r and s, the sequence defined by a(r,s;n) = A143007(r*n - 1, s*n - 1) may also satisfy the same supercongruences. This is the case r = 2, s = 1. Compare with the comments in A363864.
LINKS
Table of n, a(n) for n=0..14.
FORMULA
a(n) = Sum_{k = 0..n-1} binomial(2*n-1, k)^2 * binomial(3*n-2-k, 2*n-1)^2.
a(n) = hypergeom([2*n, 1 - 2*n, n, 1 - n], [1, 1, 1], 1).
P-recursive: 2*(n-1)^3*(2*n-1)^3*(440*n^3-2178*n^2+3600*n-1987)*a(n) = (865920*n^9 - 9481824*n^8 + 45492136*n^7 - 125359294*n^6 + 218361816*n^5 - 249018285*n^4 + 185709390*n^3 - 87271191*n^2 + 23447876*n - 2745998)*a(n-1) - 2*(2*n-3)^3*(n-2)^3*(440*n^3-858*n^2+564*n-125)*a(n-2) with a(1) = 1 and a(2) = 25.
a(n) ~ phi^(10*n - 4) / (2^(5/2) * 5^(1/4) * (Pi*n)^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 16 2023
MAPLE
seq( add(binomial(2*n-1, k)^2 * binomial(3*n-2-k, 2*n-1)^2, k = 0..n-1), n = 1..20);
# alternative program
seq(simplify(hypergeom([2*n, 1 - 2*n, n, 1 - n], [1, 1, 1], 1)), n = 1..20);
MATHEMATICA
Table[HypergeometricPFQ[{2*n, 1 - 2*n, n, 1 - n}, {1, 1, 1}, 1], {n, 1, 20}] (* Vaclav Kotesovec, Jul 16 2023 *)
CROSSREFS
Cf. A005259, A143007, A363864, A364243.
KEYWORD
nonn,easy,new
AUTHOR
Peter Bala, Jul 16 2023
STATUS
approved
A363564 Expansion of g.f. A(x) satisfying 1 = Sum_{n>=0} x^n * A(x)^(3*n) / (1 + x^(n+1)*A(x)^4). +0
0
1, 2, 15, 160, 1979, 26633, 378612, 5593669, 85036458, 1321547904, 20901013044, 335307963490, 5443261450865, 89249920538306, 1475910492040246, 24587479259900805, 412252774520658173, 6951447807236206940, 117807212665434783089, 2005490388805271264356 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The g.f. A(x) of this sequence is motivated by the following identity:
Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;
here, p = x*A(x)^3, q = -x*A(x)^4, and r = x.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1) 1 = Sum_{n>=0} x^n * A(x)^(3*n) / (1 + x^(n+1)*A(x)^4).
(2) 1 = Sum_{n>=0} (-1)^n * x^n * A(x)^(4*n) / (1 - x^(n+1)*A(x)^3).
(3) x = Sum_{n>=1} (-1)^(n-1) * x^(n^2) * A(x)^(7*(n-1)) * (1 + x^(2*n)*A(x)^7) / ((1 - x^n*A(x)^3)*(1 + x^n*A(x)^4)).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 15*x^2 + 160*x^3 + 1979*x^4 + 26633*x^5 + 378612*x^6 + 5593669*x^7 + 85036458*x^8 + 1321547904*x^9 + 20901013044*x^10 + ...
where
1 = 1/(1 + x*A(x)^4) + x*A(x)^3/(1 + x^2*A(x)^4) + x^2*A(x)^6/(1 + x^3*A(x)^4) + x^3*A(x)^9/(1 + x^4*A(x)^4) + x^4*A(x)^12/(1 + x^5*A(x)^4) + ...
also,
1 = 1/(1 - x*A(x)^3) - x*A(x)^4/(1 - x^2*A(x)^3) + x^2*A(x)^8/(1 - x^3*A(x)^3) - x^3*A(x)^12/(1 - x^4*A(x)^3) + x^4*A(x)^16/(1 - x^5*A(x)^3) -+ ...
PROG
(PARI) {a(n, k=4) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-1 + sum(n=0, #A, x^n * Ser(A)^((k-1)*n) / (1 + x^(n+1)*Ser(A)^k ) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n, 4), ", "))
CROSSREFS
Cf. A363562, A363563, A340329, A340355.
KEYWORD
nonn,new
AUTHOR
Paul D. Hanna, Jul 19 2023
STATUS
approved
A363563 Expansion of g.f. A(x) satisfying 1 = Sum_{n>=0} x^n * A(x)^(2*n) / (1 + x^(n+1)*A(x)^3). +0
0
1, 2, 11, 84, 738, 7029, 70570, 735401, 7879118, 86249454, 960434270, 10845322135, 123896322956, 1429327711980, 16628329185358, 194858230552674, 2297972689389087, 27252117638208701, 324797817830706494, 3888255542301372866, 46733817274361827340, 563736664663891455990 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The g.f. A(x) of this sequence is motivated by the following identity:
Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;
here, p = x*A(x)^2, q = -x*A(x)^3, and r = x.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1) 1 = Sum_{n>=0} x^n * A(x)^(2*n) / (1 + x^(n+1)*A(x)^3).
(2) 1 = Sum_{n>=0} (-1)^n * x^n * A(x)^(3*n) / (1 - x^(n+1)*A(x)^2).
(3) x = Sum_{n>=1} (-1)^(n-1) * x^(n^2) * A(x)^(5*(n-1)) * (1 + x^(2*n)*A(x)^5) / ((1 - x^n*A(x)^2)*(1 + x^n*A(x)^3)).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 11*x^2 + 84*x^3 + 738*x^4 + 7029*x^5 + 70570*x^6 + 735401*x^7 + 7879118*x^8 + 86249454*x^9 + 960434270*x^10 + ...
where
1 = 1/(1 + x*A(x)^3) + x*A(x)^2/(1 + x^2*A(x)^3) + x^2*A(x)^4/(1 + x^3*A(x)^3) + x^3*A(x)^6/(1 + x^4*A(x)^3) + x^4*A(x)^8/(1 + x^5*A(x)^3) + ...
also,
1 = 1/(1 - x*A(x)^2) - x*A(x)^3/(1 - x^2*A(x)^2) + x^2*A(x)^6/(1 - x^3*A(x)^2) - x^3*A(x)^9/(1 - x^4*A(x)^2) + x^4*A(x)^12/(1 - x^5*A(x)^2) -+ ...
PROG
(PARI) {a(n, k=3) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-1 + sum(n=0, #A, x^n * Ser(A)^((k-1)*n) / (1 + x^(n+1)*Ser(A)^k ) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n, 3), ", "))
CROSSREFS
Cf. A363562, A363564, A340329, A340355.
KEYWORD
nonn,new
AUTHOR
Paul D. Hanna, Jul 19 2023
STATUS
approved
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Last modified July 21 04:11 EDT 2023. Contains 363988 sequences. (Running on oeis4.)