A345235 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, 1, 2, 3, 5, 8, 14, 25, 44, 78, 142, 261, 479, 886, 1655, 3105, 5843, 11043, 20965, 39938, 76285, 146123, 280691, 540475, 1042885, 2016481, 3906647, 7582034, 14739395, 28697969, 55958110, 109262713, 213619535, 418158580, 819491034, 1607764395, 3157551026, 6207346544

Offset

1,5

Links

Table of n, a(n) for n=1..40.

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

a(n) ~ c * d^n / n^(3/2), where d = 2.04187801797233390910633071122033289228232310618876458... and c = 0.624667034123125135463988884805660643637934291759335... - Vaclav Kotesovec, Jun 19 2021

Mathematica

nmax = 40; A[_] = 0; Do[A[x_] = x + x^2 Exp[Sum[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 Sum[(-1)^d d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 40}]

Crossrefs

Cf. A007560, A007562, A045648, A345234.

Sequence in context: A104882 A091956 A107480 * A128021 A036241 A192633

Adjacent sequences: A345232 A345233 A345234 * A345236 A345237 A345238

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 11 2021

Status

approved