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A024395
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a(n) = n-th elementary symmetric function of the first n+1 positive integers congruent to 2 mod 3.
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5
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1, 7, 66, 806, 12164, 219108, 4591600, 109795600, 2951028000, 88084714400, 2891353030400, 103521905491200, 4015191638617600, 167714507921497600, 7506196028811110400, 358368551285791692800, 18180562447078051328000
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OFFSET
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0,2
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COMMENTS
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Comment by R. J. Mathar, Oct 01 2016 (Start):
The k-th elementary symmetric functions of the integers 2+j*3, j=0..n-1, form a triangle T(n,k), 0<=k<=n, n>=0:
1
1 2
1 7 10
1 15 66 80
1 26 231 806 880
1 40 595 4040 12164 12320
1 57 1275 14155 80844 219108 209440
1 77 2415 39655 363944 1835988 4591600 4188800
1 100 4186 95200 1276009 10206700 46819324 109795600 96342400
This here is the first subdiagonal. The diagonal seems to be A008544. The first columns are A000012, A005449, A024391, A024392. (End)
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
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E.g.f. (for offset 1): -(1/3)*log(1-3*x)/(1-3*x)^(2/3). - Vladeta Jovovic, Sep 26 2003
For n >= 1, a(n-1) = 3^(n-1)*n!*sum(binomial(k-1/3,k)/(n-k), k = 0..n-1). - Milan Janjic, Dec 14 2008, corrected by Peter Bala, Oct 08 2013
a(n) ~ (n+1)! * 3^n * (log(n) + gamma - Pi*sqrt(3)/6 + 3*log(3)/2) / (n^(1/3)*GAMMA(2/3)), where "GAMMA" is the Gamma function and "gamma" is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 07 2013
a(n+1) = (6*n+7) * a(n) - (3*n+2)^2 * a(n-1). - Gheorghe Coserea, Aug 30 2015
a(n) = A225470(n+1, 1), n >= 0. - Wolfdieter Lang, May 29 2017
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EXAMPLE
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From Gheorghe Coserea, Dec 24 2015: (Start)
For n=1 we have a(1) = 2*5*(1/2 + 1/5) = 7.
For n=2 we have a(2) = 2*5*8*(1/2 + 1/5 + 1/8) = 66.
For n=3 we have a(3) = 2*5*8*11*(1/2 + 1/5 + 1/8 + 1/11) = 806.
(End)
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MATHEMATICA
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Table[ (-1)^(n+1)*Sum[(-3)^(n - k) k (-1)^(n - k) StirlingS1[n+1, k + 1], {k, 0, n}], {n, 1, 30}]
Join[{1}, Table[Module[{c=NestList[3+#&, 2, n+1]}, Times@@c*Total[1/c]], {n, 0, 20}]] (* Harvey P. Dale, Jul 09 2019 *)
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PROG
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(PARI)
n = 16; a = vector(n); a[1] = 7; a[2] = 66;
for (k=2, n-1, a[k+1] = (6*k+7) * a[k] - (3*k+2)^2 * a[k-1]);
print(concat(1, a)) \\ Gheorghe Coserea, Aug 30 2015
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CROSSREFS
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Cf. A024216, A225470 (second column).
Sequence in context: A300991 A122705 A185181 * A215077 A003286 A244602
Adjacent sequences: A024392 A024393 A024394 * A024396 A024397 A024398
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling
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EXTENSIONS
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Formula (see Mathematica line), correction and more terms from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 21 2001
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STATUS
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approved
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