A199033 Number of ways to place n non-attacking bishops on a 2 X 2n board.

1, 4, 22, 128, 771, 4744, 29618, 186880, 1188679, 7608764, 48953224, 316283264, 2050706932, 13336273528, 86953633242, 568221290496, 3720529001823, 24403423540348, 160314652983158, 1054635453261568, 6946703172803003, 45809043607167328, 302395650703501688

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

V. Kotesovec, Non-attacking chess pieces

Formula

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

a(n) = Sum_{j=0..n} (binomial(2n-j+1,j)*binomial(n+j+1,n-j)).

a(n) ~ 3^(3n+4)/2^(2n+5)/sqrt(3*Pi*n).

Self-convolution of A219197. - Paul D. Hanna, Nov 14 2012

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

a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(2*n+2)). - Ilya Gutkovskiy, Oct 25 2017

Mathematica

Table[Sum[Binomial[2n-j+1, j]*Binomial[n+j+1, n-j], {j, 0, n}], {n, 0, 25}]

Prog

(PARI) {a(n)=sum(k=0, n, binomial(n+k+1, n-k)*binomial(2*n-k+1, k))}
(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
for(n=0, 25, print1(a(n), ", "))
(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 */
(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
(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

Crossrefs

Cf. A002465, A191236, A219197, A183160.

Sequence in context: A011789 A047039 A100525 * A370695 A086682 A261399

Adjacent sequences: A199030 A199031 A199032 * A199034 A199035 A199036

Keyword

nonn

Author

Vaclav Kotesovec, Nov 02 2011

Extensions

Offset changed to 0 and a(0)=1 added by Paul D. Hanna, Nov 14 2012

Status

approved