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%I #6 Mar 02 2014 16:55:30
%S 1,3,154,7941,429036,24123105,1399303662,83176893681,5041470373624,
%T 310449199290489,19369215839341710,1221826010437625703,
%U 77798300823672280164,4993767938470070592261,322795606469564782029126
%N a(0)=1; for n > 0, a(n)=sum(binomial(n,k)*(binomial(n+k,k+1)^2)* binomial(n+k,k),k=0..n).
%F Special values of the hypergeometric function 4F3, in Maple notation: a(n)=n^2*hypergeom([n+1, n+1, n+1, -n], [1, 2, 2], -1), n=1, 2....
%F Recurrence: 4*(n-1)^2*(n+1)^2*(29412*n^8 - 523944*n^7 + 3995715*n^6 - 17035566*n^5 + 44400751*n^4 - 72401280*n^3 + 72040928*n^2 - 39898368*n + 9379584)*a(n) = 2*(4176504*n^12 - 80664804*n^11 + 681787110*n^10 - 3320158377*n^9 + 10317109990*n^8 - 21386158690*n^7 + 29990437762*n^6 - 28191974977*n^5 + 17142077578*n^4 - 6206039632*n^3 + 1063468848*n^2 + 4760064*n - 18759168)*a(n-1) + 4*(823536*n^12 - 17141040*n^11 + 157530180*n^10 - 841802850*n^9 + 2898036925*n^8 - 6724482767*n^7 + 10676838689*n^6 - 11506383284*n^5 + 8122517663*n^4 - 3444627899*n^3 + 682922191*n^2 + 15710352*n - 19910592)*a(n-2) + 2*(n-3)*(235296*n^11 - 4544496*n^10 + 38143266*n^9 - 182713089*n^8 + 551659187*n^7 - 1094072109*n^6 + 1439496807*n^5 - 1235496354*n^4 + 653944224*n^3 - 183118980*n^2 + 13441600*n + 2965248)*a(n-3) - (n-4)^2*(n-3)^2*(29412*n^8 - 288648*n^7 + 1151643*n^6 - 2417028*n^5 + 2879446*n^4 - 1930604*n^3 + 642371*n^2 - 53200*n - 12768)*a(n-4). - _Vaclav Kotesovec_, Mar 02 2014
%F a(n) ~ c * d^n / n^(3/2), where d = 71.39297952064022156... is the root of the equation 1 - 16*d - 112*d^2 - 284*d^3 + 4*d^4 = 0, and c = 0.2216473197208166381284001749414... - _Vaclav Kotesovec_, Mar 02 2014
%t Flatten[{1,Table[Sum[Binomial[n,k]*(Binomial[n+k,k+1]^2)* Binomial[n+k,k],{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Mar 02 2014 *)
%Y Cf. A074635, A074649, A073530.
%K nonn
%O 0,2
%A _Karol A. Penson_, Sep 02 2002
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