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

0,3

Links

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

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

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 *)

Prog

(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
(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
(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
(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

Crossrefs

Sequence in context: A052946 A026152 A025179 * A221420 A212262 A233347

Adjacent sequences: A116385 A116386 A116387 * A116389 A116390 A116391

Keyword

easy,nonn

Author

Paul Barry, Feb 12 2006

Status

approved