A125282 G.f. satisfies: A(x) = Sum{n>=0} x^n * A(n*x).

1, 1, 2, 5, 17, 80, 525, 4839, 62936, 1158785, 30277579, 1124649526, 59465788597, 4480380804517, 481401971074410, 73812092299235769, 16158739669470307453, 5052972095683109687920, 2257981256268589345121153

Offset: 0

Author

Paul D. Hanna, Nov 29 2006

Keywords

• nonn

Examples

			A(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 80*x^5 + 525*x^6 + 4839*x^7 +...
G.f. A(x) satisfies:
A(x) = 1 + x*A(x) + x^2*A(2x) + x^3*A(3x) + x^4*A(4x) + x^5*A(5x) +...
which leads to the recurrence illustrated by:
a(4) = 4^0*(1) + 3^1*(1) + 2^2*(2) + 1^3*(5) = 17;
a(5) = 5^0*(1) + 4^1*(1) + 3^2*(2) + 2^3*(5) + 1^4*(17) = 80;
a(6) = 6^0*(1) + 5^1*(1) + 4^2*(2) + 3^3*(5) + 2^4*(17) + 1^5*(80) = 525.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..100
• Vaclav Kotesovec, Graph, asymptotic behavior

Crossrefs

Cf. A125281 (variant), A210525.

Programs

Mathematica

nmax = 20; aa = ConstantArray[0, nmax]; aa[[1]] = 1;Do[aa[[n]] = 1 + Sum[(n-k)^k*aa[[k]], {k, 1, n - 1}], {n, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Feb 23 2014 *)

PARI

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

Formula

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

Limit n->infinity (a(n))^(1/n^2) = 3^(1/6). - Vaclav Kotesovec, Feb 24 2014

a(n) ~ c * 3^(n^2/6 - n/2), where c = 372374.41350200494715367264093778... if n=3k, c = 372374.41350258936507380006951913... if n=3k+1, and c = 372374.41350254286383864609841301... if n=3k+2. - Vaclav Kotesovec, Feb 24 2014