A188365 a(n) = n! * [x^n] exp((1 - 2*x)/(1 - 3*x + x^2) - 1).

1, 1, 5, 43, 505, 7421, 130501, 2668975, 62197073, 1626103225, 47116726021, 1498191224531, 51855200633545, 1940384578283893, 78042911672096645, 3357060094366363351, 153771739817047383841, 7471843888639307665265, 383835896530177022152453, 20783664252941721959512315

Offset: 0

Author

Vladimir Kruchinin, Mar 28 2011

Keywords

• nonn

Links

• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Crossrefs

Cf. A001519, A188137.

Programs

Maple

gf := exp((1 - 2*x)/(1 - 3*x + x^2) - 1): ser := series(gf, x, 22):
seq(k!*coeff(ser, x, k), k=0..19); # Peter Luschny, Jul 30 2020

PARI

f(n,m) = sum(k=m, n, binomial(n-1,k-1) * sum(i=ceil((k-m)/2), k-m, binomial(i,k-m-i)*binomial(m+i-1,m-1))); \\ A188137
a(n) = if (n, n!*sum(k=1, n, f(n,k)/k!), 1); \\ Michel Marcus, Jul 30 2020

PARI

my(x='x+O('x^25)); Vec(serlaplace(exp((1-2*x)/(1-3*x+x^2)-1))) \\ Joerg Arndt, Jul 30 2020

Formula

E.g.f.: exp((1 - 2*x)/(1 - 3*x + x^2) - 1) = exp(G(x) - 1) where G(x) is the o.g.f. of A001519.

a(n) = n! * Sum_{k=1..n} A188137(n,k)/k!, n>0, a(0)=1.

Extensions

More terms from Michel Marcus, Jul 30 2020