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

0,4

Comments

From Peter Bala, Nov 26 2010: (Start)

Define a polynomial sequence P_n(x) recursively by

... P_0(x) = 1,

... P_n(x) = (x-1)*P_(n-1)(x-1) + n*P_(n-1)(x+1) for n >= 1.

The first few polynomials are

P_1(x) = x;

P_2(x) = x^2 + 3;

P_3(x) = x^3 + 12*x + 8.

It appears that a(n+1) = P_n(1) (checked as far as a(19)).

Compare with A173895. (End)

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

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

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 *)

Prog

(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

Crossrefs

Cf. A144011, A238302.

Sequence in context: A166901 A060072 A157503 * A366184 A179496 A339233

Adjacent sequences: A144007 A144008 A144009 * A144011 A144012 A144013

Keyword

nonn

Author

Paul D. Hanna, Sep 10 2008

Status

approved