%I #30 Oct 19 2024 21:40:39
%S 1,1,5,43,505,7421,130501,2668975,62197073,1626103225,47116726021,
%T 1498191224531,51855200633545,1940384578283893,78042911672096645,
%U 3357060094366363351,153771739817047383841,7471843888639307665265,383835896530177022152453,20783664252941721959512315
%N a(n) = n! * [x^n] exp((1 - 2*x)/(1 - 3*x + x^2) - 1).
%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F 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.
%F a(n) = n! * Sum_{k=1..n} A188137(n,k)/k!, n>0, a(0)=1.
%p gf := exp((1 - 2*x)/(1 - 3*x + x^2) - 1): ser := series(gf, x, 22):
%p seq(k!*coeff(ser, x, k), k=0..19); # _Peter Luschny_, Jul 30 2020
%o (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
%o a(n) = if (n, n!*sum(k=1, n, f(n,k)/k!), 1); \\ _Michel Marcus_, Jul 30 2020
%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(exp((1-2*x)/(1-3*x+x^2)-1))) \\ _Joerg Arndt_, Jul 30 2020
%Y Cf. A001519, A188137.
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, Mar 28 2011
%E More terms from _Michel Marcus_, Jul 30 2020