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%I #37 Jun 11 2023 12:15:18
%S 1,513,47169,1775169,37712169,534505257,5587534953,46011772521,
%T 312480135396,1809674119396,9165388162788,41395684407012,
%U 169328324418084,635173167426084,2207399512578084,7167715400927268,21902130296812161,63361228916945025,174437774859945025
%N a(n) = Sum_{i=1..n} C(i+6,7)^3.
%H Vincenzo Librandi, <a href="/A086030/b086030.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_23">Index entries for linear recurrences with constant coefficients</a>, signature (23, -253, 1771, -8855, 33649, -100947, 245157, -490314, 817190, -1144066, 1352078, -1352078, 1144066, -817190, 490314, -245157, 100947, -33649, 8855, -1771, 253, -23, 1).
%F a(n) = C(n+7,8)*(5*C(n+13,14) + 210*C(n+12,14) + 1491*C(n+11,14) + 2828*C(n+10,14) + 1491*C(n+9,14) + 210*C(n+8,14) + 5*C(n+7,14))/5. - _Yahia Kahloune_, Dec 22 2013
%F -(n-1)^3*a(n) +(2*n+5)*(n^2+5*n+43)*a(n-1) -(n+6)^3*a(n-2)=0. - _R. J. Mathar_, Dec 22 2013
%F G.f.: -x*(x^14 + 490*x^13 + 35623*x^12 + 818300*x^11 + 7917371*x^10 + 37215794*x^9 + 91789005*x^8 + 123519792*x^7 + 91789005*x^6 + 37215794*x^5 + 7917371*x^4 + 818300*x^3 + 35623*x^2 + 490*x + 1)/(x-1)^23. - _Vaclav Kotesovec_, Dec 23 2013
%F a(n) = (1/28165294080000)*n^2*(1 + n)^2*(2 + n)^2*(3 + n)^2*(4 + n)^2*(5 + n)^2*(6 + n)^2*(7 + n)^2*(-3 + 3234*n + 6979*n^2 + 5292*n^3 + 1603*n^4 + 210*n^5 + 10*n^6). - _G. C. Greubel_, Nov 22 2017
%e a(3) = Sum_{i=1..3} C(6+i,7)^3 = C(10,8)*(5*C(16,14) + 210*C(15,14) + 1491*C(14,14))/5 = 47169.
%p A086030:=n->add(binomial(i+6,7)^3, i=1..n); seq(A086030(n), n=1..30); # _Wesley Ivan Hurt_, Dec 22 2013
%t Table[Sum[Binomial[i + 6, 7]^3, {i, n}], {n, 30}] (* _Wesley Ivan Hurt_, Dec 22 2013 *)
%o (PARI) a(n) = sum(i=1, n, binomial(i+6, 7)^3); \\ _Michel Marcus_, Dec 22 2013
%o (Magma) [(1/28165294080000)*n^2*(1 + n)^2*(2 + n)^2*(3 + n)^2*(4 + n)^2*(5 + n)^2*(6 + n)^2*(7 + n)^2*(-3 + 3234*n + 6979*n^2 + 5292*n^3 + 1603*n^4 + 210*n^5 + 10*n^6): n in [1..30]]; // _G. C. Greubel_, Nov 22 2017
%Y Cf. A087127, A024166, A085438 - A085442, A086020 - A086029.
%K easy,nonn
%O 1,2
%A _André F. Labossière_, Jul 11 2003