A203265 L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} 2*a(n*k)*x^(n*k)/k ).

1, 5, 22, 125, 576, 3554, 16843, 103917, 521338, 3189600, 15813205, 101516930, 501568809, 3154939135, 16288999167, 101770328205, 513944896547, 3322082384450, 16707380500562, 106553006536680, 554390049927421, 3479202589748077, 17774723219041838

Offset

1,2

Comments

L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} G_n(x^n)^2 * x^n/n where G_n(x) = exp( Sum_{k>=1} a(n*k)*x^k/k ) are integer series.

Links

Table of n, a(n) for n=1..23.

Formula

Equals the logarithmic derivative of A203266.

Example

L.g.f.: L(x) = x + 5*x^2/2 + 22*x^3/3 + 125*x^4/4 + 576*x^5/5 + 3554*x^6/6 +...
L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} G_n(x^n)^2*x^n/n
where G_n(x) = exp( Sum_{k>=1} a(n*k)*x^k/k ), which begin:
G_1(x) = 1 + x + 3*x^2 + 10*x^3 + 43*x^4 + 172*x^5 + 852*x^6 +...
G_2(x) = 1 + 5*x + 75*x^2 + 1518*x^3 + 34663*x^4 + 867760*x^5 +...;
G_3(x) = 1 + 22*x + 2019*x^2 + 214648*x^3 + 31221037*x^4 +...;
G_4(x) = 1 + 125*x + 59771*x^2 + 40659310*x^3 + 31438395303*x^4 +...;
G_5(x) = 1 + 576*x + 1760688*x^2 + 6380121685*x^3 +...;
G_6(x) = 1 + 3554*x + 57073923*x^2 + 1295238092004*x^3 +...;
G_7(x) = 1 + 16843*x + 1719312892*x^2 + 212162358939394*x^3 +...;
G_8(x) = 1 + 103917*x + 56284535547*x^2 + 44125115136389518*x^3 +...; ...

Prog

(PARI) {a(n)=local(L=vector(n, i, 1)); for(i=1, n, L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor(n/m), 2*L[m*k]*x^(m*k)/k)+x*O(x^n)))))); L[n]}

Crossrefs

Cf. A203266 (exp), A203253, A203267.

Sequence in context: A159596 A020077 A265998 * A355886 A033462 A111154

Adjacent sequences: A203262 A203263 A203264 * A203266 A203267 A203268

Keyword

nonn

Author

Paul D. Hanna, Dec 30 2011

Status

approved