A086027 a(n) = Sum_{i=1..n} binomial(i+5,6)^2.
1, 50, 834, 7890, 51990, 265434, 1119210, 4063866, 13081875, 38131900, 102259964, 255425340, 600047436, 1336192860, 2838530460, 5783112156, 11350211925, 21540508734, 39656591950, 71021001950, 124026854850, 211648774950, 353581802550, 579225802950, 931794553575
Offset: 1
Links
- T. D. Noe, Table of n, a(n) for n = 1..1000
- John Engbers and Christopher Stocker, Two Combinatorial Proofs of Identities Involving Sums of Powers of Binomial Coefficients, Integers 16 (2016), #A58.
- Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13..
- Index entries for linear recurrences with constant coefficients, signature (14, -91, 364, -1001, 2002, -3003, 3432, -3003, 2002, -1001, 364, -91, 14, -1).
Crossrefs
Programs
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GAP
List([1..30], n-> Sum([1..n], j-> Binomial(j+5,6)^2)); # G. C. Greubel, Aug 27 2019
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Magma
[n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(77*n^6 +1386*n^5 +9380*n^4 + 29400*n^3 +41783*n^2 +20874*n +60)/518918400: n in [1..30]]; // G. C. Greubel, Nov 22 2017
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Maple
A086027:=n->sum(binomial(i+5,6)^2, i=1..n); seq(A086027(k), k=1..50); # Wesley Ivan Hurt, Oct 24 2013
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Mathematica
Table[Sum[Binomial[k + 5, 6]^2, {k, 1, n}], {n, 50}] (* Wesley Ivan Hurt, Oct 24 2013 *) -
PARI
vector(30, n, sum(i=1,n, binomial(i+5,6)^2) ) \\ G. C. Greubel, Nov 22 2017
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Sage
[sum(binomial(j+5,6)^2 for j in (1..n)) for n in (1..30)] # G. C. Greubel, Aug 27 2019
Formula
From R. J. Mathar, Jun 16 2010: (Start)
G.f.: x*(1 + 36*x + 225*x^2 + 400*x^3 + 225*x^4 + 36*x^5 + x^6)/(1-x)^14.
a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(77*n^6 + 1386*n^5 + 9380*n^4 + 29400*n^3 + 41783*n^2 + 20874*n + 60)/518918400. (End)
Extensions
More terms from R. J. Mathar, Jun 16 2010