A319365 Expansion of e.g.f. exp(x^4/4)/(1 - x).

1, 1, 2, 6, 30, 150, 900, 6300, 51660, 464940, 4649400, 51143400, 614968200, 7994586600, 111924212400, 1678863186000, 26865216378000, 456708678426000, 8220756211668000, 156194368021692000, 3123907159441068000, 65602050348262428000, 1443245107661773416000, 33194637476220788568000

Offset: 0

Author

Ilya Gutkovskiy, Sep 17 2018

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A000138, A000522, A030975, A060706, A130905, A319364.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x^4/4)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Dec 28 2018

Maple

f:= gfun:-rectoproc({(n+1)*(n+2)*(n+3)*(n+4)*a(n)-(n+2)*(n+3)*(n+4)*a(n+1)-(n+5)*a(n+4)+a(n+5)},seq(a(i)=[1,1,2,6,30][i+1],i=0..4)},a(n),remember):
map(f, [$0..30]); # Robert Israel, Dec 28 2018

Mathematica

nmax = 23; CoefficientList[Series[Exp[x^4/4]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) ~ n!*exp(1/4).

(n+1)*(n+2)*(n+3)*(n+4)*a(n)-(n+2)*(n+3)*(n+4)*a(n+1)-(n+5)*a(n+4)+a(n+5)=0. - Robert Israel, Dec 28 2018