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a(n) = Sum_{i=1..n} C(i+6,7)^2.
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%I #38 Jun 11 2023 12:14:01

%S 1,65,1361,15761,124661,751925,3696581,15475205,56884430,187758030,

%T 565982734,1578749710,4117700254,10127050654,23648089054,52733344990,

%U 112835299639,232623278455,463695768455,896396608455,1684993889355,3086944610955,5522978819355

%N a(n) = Sum_{i=1..n} C(i+6,7)^2.

%H G. C. Greubel, <a href="/A086029/b086029.txt">Table of n, a(n) for n = 1..5000</a>

%H John Engbers and Christopher Stocker, <a href="http://epublications.marquette.edu/mscs_fac/456/">Two Combinatorial Proofs of Identities Involving Sums of Powers of Binomial Coefficients</a>, Integers 16 (2016), #A58.

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (16, -120, 560, -1820, 4368, -8008, 11440, -12870, 11440, -8008, 4368, -1820, 560, -120, 16, -1).

%F 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

%F (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

%F 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

%F 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

%e 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.

%p A086029:=n->add(binomial(i+6, 7)^2, i=1..n); seq(A086029(n), n=1..30); # _Wesley Ivan Hurt_, Dec 22 2013

%t Table[Sum[Binomial[i + 6, 7]^2, {i, n}], {n, 30}] (* _Wesley Ivan Hurt_, Dec 22 2013 *)

%t 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 *)

%o (PARI) a(n) = sum(i=1, n, binomial(i+6, 7)^2); \\ _Michel Marcus_, Dec 22 2013

%o (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

%o (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

%Y Cf. A087127, A024166, A085438-A085442, A086020-A086030, A234253.

%K easy,nonn

%O 1,2

%A _André F. Labossière_, Jul 11 2003

%E More terms from _Michel Marcus_, Dec 22 2013