A086029 a(n) = Sum_{i=1..n} C(i+6,7)^2.
1, 65, 1361, 15761, 124661, 751925, 3696581, 15475205, 56884430, 187758030, 565982734, 1578749710, 4117700254, 10127050654, 23648089054, 52733344990, 112835299639, 232623278455, 463695768455, 896396608455, 1684993889355, 3086944610955, 5522978819355
Offset: 1
Examples
a(3) = C(10,8)*(-1*C(10,0) + 8*C(10,1) - 36*C(10,2) + 120*C(10,3) - 330*C(10,4) + 792*C(10,5) - 1716*C(10,6) + 3432*C(10,7))/6435 = 1361.
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 (16, -120, 560, -1820, 4368, -8008, 11440, -12870, 11440, -8008, 4368, -1820, 560, -120, 16, -1).
Programs
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Magma
[(1/108972864000)*n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7+n)*(7 + 2*n)*(60 + 64974*n + 126245*n^2 + 82467*n^3 + 23408*n^4 + 3003*n^5 + 143*n^6): n in [1..30]]; // G. C. Greubel, Nov 22 2017
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Maple
A086029:=n->add(binomial(i+6, 7)^2, i=1..n); seq(A086029(n), n=1..30); # Wesley Ivan Hurt, Dec 22 2013
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Mathematica
Table[Sum[Binomial[i + 6, 7]^2, {i, n}], {n, 30}] (* Wesley Ivan Hurt, Dec 22 2013 *) LinearRecurrence[{16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1},{1,65,1361,15761,124661,751925, 3696581, 15475205, 56884430,187758030,565982734,1578749710,4117700254, 10127050654,23648089054,52733344990},40] (* Harvey P. Dale, Apr 25 2016 *) -
PARI
a(n) = sum(i=1, n, binomial(i+6, 7)^2); \\ Michel Marcus, Dec 22 2013
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PARI
Vec(x*(x+1)*(x^6+48*x^5+393*x^4+832*x^3+393*x^2+48*x+1)/(x-1)^16 + O(x^100)) \\ Colin Barker, May 02 2014
Formula
a(n) = C(n+7,8)*(-C(n+7,0) + 8*C(n+7,1) - 36*C(n+7,2) + 120*C(n+3,7) - 330*C(n+7,4) + 792*C(n+7,5) - 1716*C(n+7,6) + 3432*C(n+7,7))/6435. - Yahia Kahloune, Dec 22 2013
(n-1)^2*a(n) +(-2*n^2-10*n-37)*a(n-1) +(n+6)^2*a(n-2)=0. - R. J. Mathar, Dec 22 2013
G.f.: x*(x+1)*(x^6 +48*x^5 +393*x^4 +832*x^3 +393*x^2 +48*x +1)/(1-x)^16. - Colin Barker, May 02 2014
a(n) = (1/108972864000)*n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7+n)*(7 + 2*n)*(60 + 64974*n + 126245*n^2 + 82467*n^3 + 23408*n^4 + 3003*n^5 + 143*n^6). - G. C. Greubel, Nov 22 2017
Extensions
More terms from Michel Marcus, Dec 22 2013