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A119363 a(n) = Sum_{k=0..n} C(n,3k)^2.

%I A119363 #18 Mar 12 2019 08:08:08
%S A119363 1,1,1,2,17,101,402,1275,3921,14114,58601,243695,950578,3537847,
%T A119363 13166791,50514102,198627921,782913717,3054480306,11824753551,
%U A119363 45823049817,178682390994,700285942731,2747647985241,10767833451954,42164261091351,165225573240651
%N A119363 a(n) = Sum_{k=0..n} C(n,3k)^2.
%C A119363 a(n) - A119364(n) = A119365(n).
%H A119363 Seiichi Manyama, <a href="/A119363/b119363.txt">Table of n, a(n) for n = 0..1000</a>
%F A119363 From _Vaclav Kotesovec_, Mar 12 2019: (Start)
%F A119363 Recurrence: (n-2)*(n-1)*n*(637*n^6 - 11466*n^5 + 84364*n^4 - 324394*n^3 + 686227*n^2 - 755060*n + 336132)*a(n) = 3*(n-2)*(n-1)*(1274*n^7 - 23569*n^6 + 180194*n^5 - 733383*n^4 + 1699606*n^3 - 2208294*n^2 + 1449504*n - 351000)*a(n-1) - 3*(n-2)*(3185*n^8 - 63700*n^7 + 539028*n^6 - 2512118*n^5 + 7020469*n^4 - 11971242*n^3 + 12050010*n^2 - 6446736*n + 1362744)*a(n-2) + (14014*n^9 - 315315*n^8 + 3072678*n^7 - 16986046*n^6 + 58535088*n^5 - 129861691*n^4 + 184326992*n^3 - 159830656*n^2 + 75517728*n - 14313456)*a(n-3) + 3*(n-3)*(3185*n^8 - 63700*n^7 + 538391*n^6 - 2501394*n^5 + 6946794*n^4 - 11707256*n^3 + 11530544*n^2 - 5915328*n + 1142208)*a(n-4) + 18*(n-4)*(n-3)*(2*n - 9)*(637*n^6 - 7644*n^5 + 36589*n^4 - 88858*n^3 + 114124*n^2 - 71840*n + 16440)*a(n-5).
%F A119363 a(n) ~ 4^n / (3*sqrt(Pi*n)). (End)
%t A119363 Table[Sum[Binomial[n,3k]^2, {k,0,n}], {n,0,30}] (* _Vaclav Kotesovec_, Mar 12 2019 *)
%t A119363 Table[HypergeometricPFQ[{1/3 - n/3, 1/3 - n/3, 2/3 - n/3, 2/3 - n/3, -n/3, -n/3}, {1/3, 1/3, 2/3, 2/3, 1}, 1], {n, 0, 30}] (* _Vaclav Kotesovec_, Mar 12 2019 *)
%Y A119363 Central coefficients of number triangle A119335.
%Y A119363 a(n) = A119335(2n, n).
%Y A119363 Cf. A024493, A119358, A139468, A139469.
%K A119363 easy,nonn
%O A119363 0,4
%A A119363 _Paul Barry_, May 16 2006
%E A119363 Edited by _N. J. A. Sloane_, Jun 12 2008