A320567 Expansion of e.g.f. exp(x) * Product_{k>=1} (1 + x^k/k!).

1, 2, 4, 11, 32, 97, 355, 1423, 5696, 23141, 108149, 559693, 2913971, 14806365, 75692999, 432849976, 2780749376, 18237870285, 115493756737, 708062095921, 4354275076517, 29539724932771, 227955214198529, 1836106089485736, 14279737884301139, 105409744347318897

Offset: 0

Author

Ilya Gutkovskiy, Oct 15 2018

Keywords

• nonn

Comments

Binomial transform of A007837.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..692
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A007837, A320566.

Programs

Maple

seq(coeff(series(factorial(n)*exp(x)*mul(1+x^k/factorial(k),k=1..n),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 15 2018
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*b(n-k), k=1..n))
    end:
a:= n-> add(b(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..27);  # Alois P. Heinz, Sep 27 2019

Mathematica

nmax = 25; CoefficientList[Series[Exp[x] Product[(1 + x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 25; CoefficientList[Series[Exp[x + Sum[Sum[(-1)^(k + 1) x^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: exp(x + Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)).

a(n) = Sum_{k=0..n} binomial(n,k)*A007837(k).