A125281 E.g.f. satisfies: A(x) = Sum{n>=0} x^n * A(n*x)/n!.

1, 1, 3, 16, 149, 2316, 59047, 2429554, 159549945, 16557985432, 2693862309131, 682199144788734, 267277518618047797, 161130714885281760100, 148762112860064623199295, 209444428223095096806228346, 447998198975235291015396393713, 1450973400598977755884988875863216

Offset: 0

Author

Paul D. Hanna, Nov 29 2006, Sep 22 2007

Keywords

• nonn

Examples

			A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 149*x^4/4! + 2316*x^5/5! +...
where
A(x) = 1 + x*A(x) + x^2*A(2*x)/2! + x^3*A(3*x)/3! + x^4*A(4*x)/4! + x^5*A(5*x)/5! +...
which leads to the recurrence illustrated by:
a(3) = 1*3^0*(1) + 3*2^1*(1) + 3*1^2*(3) = 16;
a(4) = 1*4^0*(1) + 4*3^1*(1) + 6*2^2*(3) + 4*1^3*(16) = 149;
a(5) = 1*5^0*(1) + 5*4^1*(1) + 10*3^2*(3) + 10*2^3*(16) + 5*1^4*(149) = 2316.
		

Crossrefs

Cf. A230323, A125282, A218683.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n,k] * (n-k)^k * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Jul 03 2025 *)

PARI

{a(n)=if(n==0,1,sum(k=0,n-1,binomial(n,k)*(n-k)^k*a(k)))}

PARI

{a(n)=local(A=1);for(i=1,n,A=sum(k=0,n,x^k/k!*subst(A,x,k*x)+x*O(x^n)));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n-1} C(n,k)*(n-k)^k * a(k) for n>0 with a(0)=1.