A307725 G.f. A(x) satisfies: A(x) = x*exp(Sum_{n>=1} Sum_{k>=1} (-1)^(k+1)*n^k*a(n)^k*x^(n*k)/k).
0, 1, 1, 2, 8, 38, 234, 1670, 13730, 126050, 1286506, 14374806, 174922742, 2299332974, 32498831162, 491302184254, 7913576956058, 135291701108082, 2447171221364738, 46693007367175606, 937331324424610142, 19748487304680389214, 435735970210393888898, 10048153760813576981702
Offset: 0
Keywords
Examples
G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 38*x^5 + 234*x^6 + 1670*x^7 + 13730*x^8 + 126050*x^9 + ...
Programs
-
Mathematica
a[n_] := a[n] = SeriesCoefficient[x Exp[Sum[Sum[(-1)^(k + 1) j^k a[j]^k x^(j k)/k, {k, 1, n - 1}], {j, 1, n - 1}]], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 23}] a[n_] := a[n] = SeriesCoefficient[x Product[(1 + k a[k] x^k), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 23}]
Formula
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + n*a(n)*x^n).
Recurrence: a(n+1) = -(1/n) * Sum_{k=1..n} ( Sum_{d|k} d*(-d*a(d))^(k/d) ) * a(n-k+1).