A110521 Expansion of 1/(1 - (x + x^2)c(3x)), c(x) the g.f. of A000108.

1, 1, 5, 30, 224, 1871, 16771, 157668, 1533970, 15314626, 156008660, 1615147014, 16944659846, 179746651907, 1924700759635, 20776060271760, 225838715259574, 2469974866825150, 27160344857205806, 300101157823582668

Offset: 0

Author

Paul Barry, Jul 24 2005

Keywords

• easy
• nonn

Comments

Diagonal sums of A110519.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..925

Programs

Mathematica

T[0, 0] := 1; T[n_, k_] := Sum[j*3^(n - k - j)*Binomial[2*n - 2*k - j - 1, n - k - j]*Binomial[j, k]/(n - k), {j, 0, n - k}]; Table[Sum[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, Aug 29 2017 *)

PARI

concat([1], for(n=1,20, print1(sum(k=0, n\2, sum(j=0, n-k, j*binomial(2*n-2*k-j-1, n-k-j)*binomial(j, k)*3^(n-k-j)/(n-k))), ", "))) \\ G. C. Greubel, Aug 29 2017

Formula

a(0) = 1, a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..(n-k)} j*C(2*n-2*k-j-1, n-k-j)*C(j, k)*3^(n-k-j)/(n-k), n > 0.

Conjecture: 2*n*a(n) + 2*(16-11n)*a(n-1) + 2*(42-11n)*a(n-2) + (32-21n)*a(n-3) + 5*(20-7n)*a(n-4) + 6*(7-2n)*a(n-5) = 0. - R. J. Mathar, Dec 10 2011