A054443 Third convolution of A001405 (central binomial numbers).
1, 4, 14, 40, 109, 276, 682, 1624, 3810, 8744, 19868, 44496, 98941, 217780, 476786, 1036024, 2241814, 4823160, 10342180, 22076080, 46994386, 99673224, 210923364, 445000560, 937051684, 1968204496, 4127285688, 8636324768, 18045851165, 37638105588, 78404375362
Offset: 0
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
Programs
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PARI
{a(n)=local(k);if(n<0, 0, k=n\2; if(n%2, (k+4)*4^(k+2)-(k+3)*binomial(2*(k+3),k+3), (2*k+7)*4^(k+1)-binomial(2*(k+2),k+2)*(4*k+9)/2 ))}
Formula
a(2*k) = (2*k+7)*4^(k+1)-binomial(2*(k+2), k+2)*(4*k+9)/2, a(2*k+1) = (k+4)*4^(k+2)-(k+3)*binomial(2*(k+3), k+3), k >= 0.
a(n) = A054336(n+3, 3) (fourth column of convolution triangle). G.f.: (1/(1-x-x^2*c(x^2)))^4, with c(x) the g.f. for the Catalan numbers A000108.
G.f.: (c(x/(2x-1))/(1-2x))^4. - Michael Somos, Jul 31 2005