Search: keyword:new
|
|
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
|
|
|
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
|
|
|
KEYWORD
|
sign,new
|
|
AUTHOR
|
|
|
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
|
|
|
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
|
|
|
KEYWORD
|
sign,new
|
|
AUTHOR
|
|
|
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
|
|
|
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
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
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
|
|
|
LINKS
|
|
|
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
|
|
|
KEYWORD
|
nonn,easy,new
|
|
AUTHOR
|
|
|
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.
|
|
LINKS
|
|
|
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
|
|
|
KEYWORD
|
nonn,easy,new
|
|
AUTHOR
|
|
|
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
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn,easy,new
|
|
AUTHOR
|
|
|
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
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn,easy,new
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
|
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn,easy,new
|
|
AUTHOR
|
|
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn,new
|
|
AUTHOR
|
|
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn,new
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
Search completed in 0.080 seconds
|