A345234 G.f. A(x) satisfies: A(x) = x + x^2 * exp(A(x) - A(-x^2)/2 + A(x^3)/3 - A(-x^4)/4 + ...).

1, 1, 1, 2, 3, 5, 9, 17, 31, 58, 112, 218, 427, 844, 1683, 3381, 6824, 13842, 28226, 57796, 118762, 244874, 506515, 1050688, 2185095, 4555217, 9517423, 19926174, 41798031, 87833877, 184881588, 389765182, 822901122, 1739763655, 3682955618, 7806103024, 16564348106, 35187631009

Offset

1,4

Links

Table of n, a(n) for n=1..38.

Formula

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

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

a(n) ~ c * d^n / n^(3/2), where d = 2.21094707842288180828190718521597733363607957468229824761... and c = 0.664585976397397791197984310778764361056468131968... - Vaclav Kotesovec, Jun 19 2021

Mathematica

nmax = 38; A[_] = 0; Do[A[x_] = x + x^2 Exp[Sum[(-1)^(k + 1) A[(-1)^(k + 1) 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] = (1/(n - 2)) Sum[(-1)^(k + 1) Sum[(-1)^(k/d + d) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 38}]

Crossrefs

Cf. A007560, A007562, A049075, A345235.

Sequence in context: A213709 A054187 A014743 * A351360 A080889 A049858

Adjacent sequences: A345231 A345232 A345233 * A345235 A345236 A345237

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 11 2021

Status

approved