A203253 L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} a(n*k)*x^(n*k)/k ).

1, 3, 7, 23, 51, 195, 435, 1631, 4165, 14563, 34761, 141479, 327471, 1222287, 3267177, 11804959, 28562075, 114349947, 272702593, 1056583023, 2786781123, 9966908779, 24678676437, 101422669199, 243331437901, 915276550503, 2464145600011, 9064045943983, 22324762587821

Offset: 1

Author

Paul D. Hanna, Dec 30 2011

Keywords

• nonn

Comments

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

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 23*x^4/4 + 51*x^5/5 + 195*x^6/6 +...
L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} G_n(x^n)*x^n/n
where G_n(x) = exp( Sum_{k>=1} a(n*k)*x^k/k ), which begin:
G_1(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 22*x^5 + 62*x^6 + 146*x^7 +...
G_2(x) = 1 + 3*x + 16*x^2 + 104*x^3 + 724*x^4 + 5428*x^5 + 44080*x^6 +...;
G_3(x) = 1 + 7*x + 122*x^2 + 2128*x^3 + 52330*x^4 + 1109386*x^5 +...;
G_4(x) = 1 + 23*x + 1080*x^2 + 67944*x^3 + 4595792*x^4 +...;
G_5(x) = 1 + 51*x + 8582*x^2 + 1482524*x^3 + 355949360*x^4 +...;
G_6(x) = 1 + 195*x + 89752*x^2 + 53146664*x^3 + 36695632888*x^4 +...;
G_7(x) = 1 + 435*x + 705756*x^2 + 1208493276*x^3 +...;
G_8(x) = 1 + 1631*x + 7232560*x^2 + 44157620896*x^3 ...; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A203254, A209397.

Programs

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),L[m*k]*x^(m*k)/k)+x*O(x^n))))));L[n]}

Formula

Equals the logarithmic derivative of A203254.