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%I #27 Feb 04 2023 20:46:53
%S 1,2,184758,60090032,139541849878,94278969044262,126648421364527548,
%T 111019250117021378442,125257104438594491956518,
%U 121088185204450642433930072,128442558588779813655233443038,128767440665677943753184267342902
%N a(n) = Sum_{k=0..n} binomial(10*n,10*k).
%H Seiichi Manyama, <a href="/A070833/b070833.txt">Table of n, a(n) for n = 0..332</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (522,587797,-75135226,-392963125,3200000).
%F a(n) = 1/10*1024^n+1/5*(-625/2+275/2*sqrt(5))^n+1/5*(-625/2-275/2*sqrt(5))^n+1/5*(123/2+55/2*sqrt(5))^n+1/5*(123/2-55/2*sqrt(5))^n.
%F G.f.: (1 - 520*x - 404083*x^2 + 37605988*x^3 + 117888625*x^4 - 320000*x^5) / ((1 - 1024*x)*(1 - 123*x + x^2)*(1 + 625*x + 3125*x^2)). - _Colin Barker_, Mar 15 2019
%F a(2*n) = (2^(20*n-1) + Lucas(20*n) + 5^(5*n)*Lucas(10*n))/5, for n>0 and for Lucas(n) = A000032(n). - _Greg Dresden_, Feb 04 2023
%o (PARI) a(n) = sum(k=0, n, binomial(10*n, 10*k)); \\ _Michel Marcus_, Mar 15 2019
%o (PARI) Vec((1 - 520*x - 404083*x^2 + 37605988*x^3 + 117888625*x^4 - 320000*x^5) / ((1 - 1024*x)*(1 - 123*x + x^2)*(1 + 625*x + 3125*x^2)) + O(x^15)) \\ _Colin Barker_, Mar 15 2019
%Y Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), this sequence (b=10). Cf. A000032.
%K easy,nonn
%O 0,2
%A Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002
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