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
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