A345200 G.f. A(x) satisfies: A(x) = x + x^2 * exp(2 * Sum_{k>=1} A(x^k) / k).
1, 1, 2, 5, 12, 32, 84, 234, 652, 1872, 5416, 15922, 47188, 141283, 425910, 1293105, 3948080, 12118619, 37367694, 115708111, 359623780, 1121543440, 3508533500, 11006973980, 34620982004, 109157354769, 344928572562, 1092190467567, 3464955417200, 11012117992012
Offset: 1
Keywords
Programs
-
Mathematica
nmax = 30; A[] = 0; Do[A[x] = x + x^2 Exp[2 Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest a[1] = a[2] = 1; a[n_] := a[n] = (2/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 30}]
Formula
G.f.: x + x^2 / Product_{n>=1} (1 - x^n)^(2*a(n)).
a(n+2) = (2/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+2).
a(n) ~ c * d^n / n^(3/2), where d = 3.3437762102302517833309792925121217026126033230718263962128740290952197... and c = 0.3397354606156870289877990463189432389789387070060129709272911771... - Vaclav Kotesovec, Jun 19 2021