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 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 ). 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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. LINKS Paul D. Hanna, Table of n, a(n) for n = 1..100 FORMULA Equals the logarithmic derivative of A203254. EXAMPLE 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 ...; ... 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), L[m*k]*x^(m*k)/k)+x*O(x^n)))))); L[n]} CROSSREFS Cf. A203254, A209397. Sequence in context: A002715 A112052 A205491 * A179491 A219167 A293466 Adjacent sequences: A203250 A203251 A203252 * A203254 A203255 A203256 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 30 2011 STATUS approved

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Last modified September 18 11:22 EDT 2024. Contains 375999 sequences. (Running on oeis4.)