A144010 E.g.f. satisfies: A'(x) = 1/(1 - x*A(x)) with A(0)=1.
1, 1, 1, 4, 21, 160, 1525, 17760, 243145, 3833600, 68373225, 1361264000, 29925477725, 719991897600, 18817847565725, 530921477363200, 16082605690148625, 520603130117939200, 17934634668874889425
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..400
Programs
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Mathematica
FindRoot[Sqrt[Pi/2]*s*E^(-s^2/2)*(Erfi[1/Sqrt[2]]-Erfi[s/Sqrt[2]]) == -1,{s,1},WorkingPrecision->50] (* program for numerical value of the constant s, Vaclav Kotesovec, Feb 23 2014 *) a[0] = 1; a[1] = 1; a[n_] := a[n] = Sum[k Binomial[n-1, k] a[k] a[n-k-1], {k, 1, n-1}]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, May 17 2016 *) -
PARI
{a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1/(1-x*A+x*O(x^n)) )); n!*polcoeff(A, n)} for(n=0,30,print1(a(n),", ")) -
PARI
/* from Peter Bala's Formula */ {a(n)=local(P=1);if(n>=0&n<2,1,for(k=1,n-1,P=(x-1)*subst(P,x,x-1) + k*subst(P,x,x+1)));subst(P,x,1)} for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Jun 15 2013
Formula
E.g.f. satisfies: A(x) = 1 + Integral 1/(1 - x*A(x)) dx.
a(n) ~ n^(n-1) * s^n / exp(n), where s = 2.0832144900084392272885741721727173082215... is the root of the equation sqrt(Pi/2)*s*exp(-s^2/2)*(erfi(1/sqrt(2)) - erfi(s/sqrt(2))) = -1. - Vaclav Kotesovec, Feb 23 2014
a(0) = 1, a(1) = 1, a(n) = Sum_{0 < k < n} k * binomial(n-1, k) * a(k) * a(n-k-1). - Vladimir Reshetnikov, May 17 2016
Comments