A134757 A007318 * A100071.

1, 3, 11, 37, 123, 401, 1293, 4131, 13107, 41353, 129873, 406319, 1267093, 3940431, 12224579, 37845117, 116944371, 360771417, 1111332129, 3418840431, 10504903809, 32242682787, 98863833159, 302863592073, 927025884477, 2835306153351, 8665554849903

Offset: 1

Author

Gary W. Adamson, Nov 08 2007

Keywords

• nonn

Comments

Also A007318^(-1) * A037965. - Gary W. Adamson, Nov 10 2007

Examples

			a(3) = 11 = (1, 2, 1) dot (1, 2, 6) = (1 + 4 + 6), where A100071 = (1, 2, 6, 12, 30, ...).
a(3) = 11 = (1, -2, 1) dot (1, 4, 18) = (1 - 8 + 18). - _Gary W. Adamson_, Nov 10 2007
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A037965, A100071, A107965.

Programs

Magma

A134757:= func< n | (&+[(-1)^(n-k-1)*(k+1)^2*Binomial(n-1,k)*Catalan(k) : k in [0..n-1]]) >;
[A134757(n): n in [1..40]]; // G. C. Greubel, May 28 2024

Mathematica

a[n_]:= a[n]= Sum[(-1)^(n-k-1)*Binomial[n-1,k]*(k+1)*Binomial[2*k, k], {k,0,n-1}];
Table[a[n], {n,40}] (* G. C. Greubel, May 28 2024 *)

SageMath

def A134757(n): return sum((-1)^(n-k-1)*(k+1)*binomial(n-1,k)*binomial( 2*k, k) for k in range(n))
[A134757(n) for n in range(1,41)] # G. C. Greubel, May 28 2024

Formula

Binomial transform of A100071 starting [1, 2, 6, 12, 30, ...].

Inverse binomial transform of A037965 starting [1, 4, 18, 80, 350, ...].

a(n) = (n-1)! * [x^(n-1)] exp(x)*((1 + 2*x)*BesselI(0, 2*x) + 2*x*BesselI(1, 2*x)) for n>0, a(0) = 0. - Peter Luschny, Aug 26 2012

D-finite with recurrence (n-1)*a(n) = 3*(n-1)*a(n-1) +(n+1)*a(n-2) -3*(n-3)*a(n-3). - R. J. Mathar, Nov 09 2021

G.f.: x*(1-x)/((1-3*x)*sqrt((1+x)*(1-3*x))). - G. C. Greubel, May 28 2024