A116387 Expansion of 1/(sqrt(1-2*x-3*x^2)*(2-M(x))), where M(x) is the g.f. of the Motzkin numbers A001006.

1, 2, 7, 22, 72, 234, 763, 2486, 8099, 26372, 85833, 279226, 907946, 2951066, 9587981, 31140034, 101104048, 328162170, 1064856217, 3454513274, 11204337056, 36332719182, 117795920249, 381848062066, 1237615088203, 4010710218384

Offset

0,2

Comments

Binomial transform of A116383.

The substitution x-> x/(1+x+x^2) in the g.f. (this might be called an inverse Motzkin transform) yields the g.f. of A074331. - R. J. Mathar, Nov 10 2008

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

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

Conjecture: n*(17*n-142)*a(n) + (17*n^2 + 95*n + 138)*a(n-1) + (-391*n^2 + 2488*n - 2908)*a(n-2) + (-17*n^2 - 603*n + 1892)*a(n-3) + 2*(697*n-2021)*(n-4)*a(n-4) + 60*(17*n-47)*(n-4)*a(n-5) = 0. - R. J. Mathar, Nov 15 2011

a(n) ~ (1+sqrt(5))^n * (5+sqrt(5)) / 10. - Vaclav Kotesovec, Feb 08 2014

Mathematica

Table[Sum[Binomial[n, j-k]Binomial[j, n-j], {k, 0, n}, {j, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Feb 08 2012 *)

Prog

(PARI) {a(n) = sum(k=0, n, sum(j=0, n, binomial(n, j-k)*binomial(j, n-j)))}; \\ G. C. Greubel, May 23 2019
(Magma) [(&+[ (&+[Binomial(n, j-k)*Binomial(j, n-j): j in [0..n]]) : k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 23 2019
(Sage) [sum( sum(binomial(n, j-k)*binomial(j, n-j) for j in (0..n)) for k in (0..n)) 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, j-k)*Binomial(j, n-j) ))) # G. C. Greubel, May 23 2019

Crossrefs

Sequence in context: A289592 A292230 A162770 * A337805 A294006 A322573

Adjacent sequences: A116384 A116385 A116386 * A116388 A116389 A116390

Keyword

easy,nonn

Author

Paul Barry, Feb 12 2006

Status

approved