A345245 G.f. A(x) satisfies: A(x) = x + x^2 * exp(4 * Sum_{k>=1} (-1)^(k+1) * A(x^k) / k).

1, 1, 4, 10, 36, 135, 504, 2000, 8072, 33099, 138132, 582930, 2485412, 10692219, 46340984, 202175344, 887175352, 3913032212, 17338327848, 77141235796, 344491008296, 1543591834950, 6937783312048, 31270131096820, 141305878384704, 640065923118435, 2905664234243052, 13217615913137250

Offset

1,3

Links

Table of n, a(n) for n=1..28.

Formula

G.f.: x + x^2 * Product_{n>=1} (1 + x^n)^(4*a(n)).

a(n+2) = (4/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * d * a(d) ) * a(n-k+2).

Mathematica

nmax = 28; A[_] = 0; Do[A[x_] = x + x^2 Exp[4 Sum[(-1)^(k + 1) 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] = (4/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 28}]

Crossrefs

Cf. A007560, A052772, A345243, A345244.

Sequence in context: A149184 A149185 A149186 * A197552 A052572 A079725

Adjacent sequences: A345242 A345243 A345244 * A345246 A345247 A345248

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 11 2021

Status

approved