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%I #32 Sep 08 2022 08:46:00
%S 1,4,22,128,771,4744,29618,186880,1188679,7608764,48953224,316283264,
%T 2050706932,13336273528,86953633242,568221290496,3720529001823,
%U 24403423540348,160314652983158,1054635453261568,6946703172803003,45809043607167328,302395650703501688
%N Number of ways to place n non-attacking bishops on a 2 X 2n board.
%H G. C. Greubel, <a href="/A199033/b199033.txt">Table of n, a(n) for n = 0..1000</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>
%F Recurrence: (112*n^4 + 968*n^3 + 3048*n^2 + 4136*n + 2040)*a(n+2) = (728*n^4 + 5914*n^3 + 17550*n^2 + 22510*n + 10530)*a(n+1) + (189*n^4 + 1539*n^3 + 4578*n^2 + 5886*n + 2760)*a(n). - _Vaclav Kotesovec_, Oct 30 2011
%F a(n) = Sum_{j=0..n} (binomial(2n-j+1,j)*binomial(n+j+1,n-j)).
%F a(n) ~ 3^(3n+4)/2^(2n+5)/sqrt(3*Pi*n).
%F Self-convolution of A219197. - _Paul D. Hanna_, Nov 14 2012
%F G.f.: A(x) = G(x)^2 / (1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - _Paul D. Hanna_, Nov 14 2012
%F a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(2*n+2)). - _Ilya Gutkovskiy_, Oct 25 2017
%t Table[Sum[Binomial[2n-j+1,j]*Binomial[n+j+1,n-j],{j,0,n}],{n,0,25}]
%o (PARI) {a(n)=sum(k=0, n, binomial(n+k+1, n-k)*binomial(2*n-k+1, k))}
%o (PARI) {a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(G^2/(1-2*x*G^2-3*x^2*G^4), n)} \\ _Paul D. Hanna_, Nov 14 2012
%o for(n=0,25,print1(a(n),", "))
%o (Maxima) A199033(n):=sum(binomial(n+k+1, n-k)*binomial(2*n-k+1,k),k,0,n)$ makelist(A199033(n),n,0,22); /* _Martin Ettl_, Nov 15 2012 */
%o (Magma) [(&+[Binomial(2*n-j+1,j)*Binomial(n+j+1,n-j): j in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Feb 19 2019
%o (Sage) [sum(binomial(2*n-j+1,j)*binomial(n+j+1,n-j) for j in (0..n)) for n in (0..30)] # _G. C. Greubel_, Feb 19 2019
%Y Cf. A002465, A191236, A219197, A183160.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Nov 02 2011
%E Offset changed to 0 and a(0)=1 added by _Paul D. Hanna_, Nov 14 2012
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