A068100 a(1) = 1; a(n+1) = n!*Sum_{k|n} a(k)/k!.

1, 1, 3, 9, 45, 165, 1605, 6645, 82245, 626565, 7430565, 47347365, 1294747365, 7521768165, 166051339365, 2617940779365, 45296384587365, 400983812683365, 16124008954507365, 137769109363339365, 6031874478387211365, 98938319915632651365, 1994173630452264715365, 27846190369337241355365, 1807180003160859492619365

Offset

1,3

Links

Table of n, a(n) for n=1..25.

Formula

E.g.f.: Sum_{n>=1} a(n) * x^n / (n!*(1 - x^n)) = Sum_{n>=1} a(n+1) * x^n / n!. - Paul D. Hanna, Sep 04 2014

E.g.f. A(x) satisfies: d/dx A(x) = 1 + A(x) + A(x^2) + A(x^3) + ... - Ilya Gutkovskiy, May 10 2019

Example

E.g.f.: A(x) = x/(1-x) + x^2/(2!*(1-x^2)) + 3*x^3/(3!*(1-x^3)) + 9*x^4/(4!*(1-x^4)) + 45*x^5/(5!*(1-x^5)) + 165*x^6/(6!*(1-x^6)) + ... + a(n)*x^n/(n!*(1-x^n)) + ...
such that A(x) = x + 3*x^2/2! + 9*x^3/3! + 45*x^4/4! + 165*x^5/5! + 1605*x^6/6! + ... + a(n+1)*x^n/n! + ...

Mathematica

a[1] = 1; a[n_] := a[n] = (n-1)!*Sum[a[k]/k!, {k, Divisors[n-1]}]; Table[a[n], {n, 1, 25}] (* Vaclav Kotesovec, Apr 26 2020 *)

Prog

(PARI) {a(n)=if(n==1, 1, (n-1)!*sumdiv(n-1, d, a(d)/d!))}
for(n=1, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 04 2014
(PARI) /* From e.g.f.: */
{a(n)=my(A=x); if(n==1, 1, for(i=1, n, A = sum(k=1, n-1, a(k)*x^k/(k!*(1-x^k +x*O(x^n) )))); (n-1)!*polcoeff(A, n-1))}
for(n=1, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 04 2014
(PARI) N=33; v=vector(N); v[1]=1; for(n=1, N-1, v[n+1]=n!*sumdiv(n, k, v[k]/k!)); v \\ Joerg Arndt, Sep 04 2014

Crossrefs

Sequence in context: A352797 A192891 A364296 * A327648 A262129 A012821

Adjacent sequences: A068097 A068098 A068099 * A068101 A068102 A068103

Keyword

nonn

Author

Leroy Quet, Mar 22 2002

Status

approved