A116388 Expansion of 1/((1+x*(1-M(x)))*sqrt(1-2*x-3*x^2)), M(x) the g.f. of A001006.

1, 1, 4, 10, 29, 82, 236, 681, 1975, 5745, 16757, 48982, 143442, 420721, 1235663, 3633453, 10695292, 31511524, 92919758, 274203662, 809719718, 2392579638, 7073684393, 20924387460, 61925598216, 183350728661, 543095661673

Offset: 0

Author

Paul Barry, Feb 12 2006

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Programs

GAP

List([0..30], n-> Sum([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-k-j) ))); # G. C. Greubel, May 23 2019

Magma

[(&+[ (&+[Binomial(n-k, j-k)*Binomial(j, n-k-j): j in [0..n-k]]) : k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 23 2019

Mathematica

Table[Sum[Sum[Binomial[n-k,j-k]*Binomial[j,n-k-j], {j,0,n-k}], {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, May 23 2019 *)

PARI

{a(n) = sum(k=0,n\2, sum(j=0,n-k, binomial(n-k, j-k)*binomial(j,n-k-j)))}; \\ G. C. Greubel, May 23 2019

Sage

[sum( sum(binomial(n-k, j-k)*binomial(j,n-k-j) for j in (0..n)) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, May 23 2019

Formula

G.f.: 2*x/(sqrt(1-2*x-3*x^2)*(sqrt(1-2*x-3*x^2) -1 +2*x +3*x^2)).

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(n-k,j-k)*C(j,n-k-j).

Conjecture: n*a(n) + 7*(-n+1)*a(n-1) + 2*(4*n-9)*a(n-2) + (25*n-58)*a(n-3) + (-18*n+65)*a(n-4) + (-52*n+199)*a(n-5) + (-31*n+135)*a(n-6) + 6*(-n+5)*a(n-7) = 0. - R. J. Mathar, Jun 22 2016