A263895 Expansion of e.g.f.: exp(-x)*x/(1-2*x)^2.

0, 1, 6, 51, 524, 6405, 90834, 1467319, 26607384, 535277961, 11832460190, 285105945531, 7437546405156, 208846303056781, 6280859188226154, 201420656725873215, 6861297209758777904, 247422535745846839569, 9416194788956228294454, 377156775060354873848131

Offset: 0

Author

Vladimir Reshetnikov, Oct 28 2015

Keywords

• nonn

Examples

			G.f. = x + 6*x^2 + 51*x^3 + 524*x^4 + 6405*x^5 + 90834*x^6 + 1467319*x^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..401
• Eric Weisstein's MathWorld, Incomplete Gamma Function.

Crossrefs

Cf. A000354.

Programs

GAP

a:=[0,1,6];; for n in [4..25] do a[n]:=2*n*a[n-1]+3*a[n-2]+2*(3-n)*a[n-3]; od; a; # Muniru A Asiru, Jul 31 2018

Maple

seq(coeff(series(factorial(n)*exp(-x)*x/(1-2*x)^2, x,n+1),x,n),n=0..25); # Muniru A Asiru, Jul 31 2018

Mathematica

Round@Table[2^n (2 Gamma[n+2, -1/2] - Gamma[n+1, -1/2])/(4 Sqrt[E]), {n, 0, 20}]
With[{nmax = 30}, CoefficientList[Series[Exp[-x]*x/(1 - 2*x)^2, {x, 0, nmax}], x]] (* G. C. Greubel, Jul 31 2018 *)

PARI

x='x+O('x^30); concat([0], Vec(serlaplace(exp(-x)*x/(1-2*x)^2))) \\ G. C. Greubel, Jul 31 2018

Formula

a(n) = 2^n*(2*Gamma(n+2,-1/2)-Gamma(n+1,-1/2))*exp(-1/2)/4, where Gamma(a,x) is the upper incomplete Gamma function.

a(n) = (A000354(n+1)-A000354(n))/4.

For n > 0, a(n) = (-1)^(n+1)*n*hypergeom([2,1-n],[],2).

Recurrence: a(0) = 0, a(1) = 1, a(2) = 6, a(n) = 2*(n+1)*a(n-1) + 3*a(n-2) + 2*(2-n)*a(n-3).

0 = a(n)*(+2*a(n+1) - 3*a(n+2) - 8*a(n+3) + a(n+4)) + a(n+1)*(+3*a(n+1) + 4*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2) + 2*a(n+3) - a(n+4)) + a(n+3)*(+a(n+3)) for all n>=0. - Michael Somos, Oct 29 2015