login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A144010 E.g.f. satisfies: A'(x) = 1/(1 - x*A(x)) with A(0)=1. 4
1, 1, 1, 4, 21, 160, 1525, 17760, 243145, 3833600, 68373225, 1361264000, 29925477725, 719991897600, 18817847565725, 530921477363200, 16082605690148625, 520603130117939200, 17934634668874889425 (list; graph; refs; listen; history; text; internal format)
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 * A179496 A339233 A107872

Adjacent sequences: A144007 A144008 A144009 * A144011 A144012 A144013

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 10 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 28 09:43 EST 2022. Contains 358407 sequences. (Running on oeis4.)