A086023 a(n) = Sum_{i=1..n} C(i+3,4)^2.
1, 26, 251, 1476, 6376, 22252, 66352, 175252, 420277, 931502, 1933503, 3796728, 7109128, 12773528, 22137128, 37160504, 60634529, 96454754, 149963979, 228375004, 341286880, 501309380, 724811880, 1032814380, 1452040005, 2016150006, 2767184031, 3757230256
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
- 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 (10,-45,120,-210,252,-210,120,-45,10,-1).
Crossrefs
Programs
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Magma
[n*(n+1)*(n+2)*(n+3)*(n+4)*(35*n^4 +280*n^3 +685*n^2 +500*n +12 )/181440: n in [1..30]]; // G. C. Greubel, Nov 22 2017
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Mathematica
Table[n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(35*n^4 + 280*n^3 + 685*n^2 + 500*n + 12)/181440, {n, 1, 50}] (* G. C. Greubel, Nov 22 2017 *) Accumulate[Binomial[Range[30]+3,4]^2] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,26,251,1476,6376,22252,66352,175252,420277,931502},30] (* Harvey P. Dale, May 06 2018 *) -
PARI
a(n) = sum(i=1, n, binomial(i+3, 4)^2); \\ Michel Marcus, Sep 05 2013
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PARI
Vec(x*(x^4+16*x^3+36*x^2+16*x+1)/(x-1)^10 + O(x^100)) \\ Colin Barker, May 02 2014
Formula
a(n) = ( C(n+4,5)/126 )*( 126 +420*C(n-1,1) +540*C(n-1,2) +315*C(n-1,3) +70*C(n-1,4) ).
a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(35*n^4 +280*n^3 +685*n^2 +500*n +12 )/181440. - Bruno Berselli, Sep 05 2013
G.f.: x*(x^4+16*x^3+36*x^2+16*x+1) / (x-1)^10. - Colin Barker, May 02 2014
Extensions
More terms from Michel Marcus, Sep 05 2013