A153732 Binomial transform of A109747.

1, 3, 8, 19, 41, 84, 171, 347, 690, 1385, 2825, 5438, 11077, 24535, 33720, 102623, 350605, -1120228, 5876775, 11232063, -256532422, 1748895117, -4057110163, -42841409122, 605093026361, -3691581277925, 3538657621384, 186391745956155, -2296017574506751

Offset: 0

Author

Gary W. Adamson, Dec 31 2008

Keywords

• sign

Comments

Equals triple binomial transform of A014182.

Examples

			a(3) = 19 = (1, 3, 3, 1) dot (1, 2, 3, 3) = (1 + 6 + 9 + 3); where A109747 = (1, 2, 3, 3, 2, 3, 5, -4, 5, 55, -212, ...).
		

Links

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

Crossrefs

Cf. A109747, A014182, A126617.

Programs

Mathematica

Join[{1}, Rest[CoefficientList[Series[Exp[2*x + 1 - Exp[-x]], {x, 0, 50}], x]*Range[0, 50]!]] (* G. C. Greubel, Aug 31 2016 *)

Formula

A007318 * A109747.

E.g.f.: exp(2*x+1-exp(-x)) = 1+3*x+8*x^2/2!+19*x^3/3!+....

a(n) = exp(1)*Sum_{k >= 0} (-1)^k*(2-k)^n/k!. Cf. A126617. - Peter Bala, Oct 28 2011.

G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1+k*x-2*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 17 2013

a(0) = 1; a(n) = 2*a(n-1) - Sum_{k=1..n} (-1)^k * binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Dec 01 2023