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

Author

Paul Barry, Feb 12 2006

Keywords

• easy
• nonn

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

Programs

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

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

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 *)

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

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

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