A346020 G.f. A(x) satisfies: A(x) = x + x^3 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

1, 0, 1, 1, 1, 2, 3, 4, 7, 11, 17, 28, 46, 74, 124, 206, 343, 577, 976, 1649, 2808, 4792, 8200, 14073, 24228, 41782, 72246, 125164, 217262, 377784, 658072, 1148006, 2005743, 3509125, 6147422, 10782375, 18934209, 33285291, 58575080, 103181405, 181928014, 321059155

Offset

1,6

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..3800

Formula

G.f.: x + x^3 / Product_{n>=1} (1 - x^n)^a(n).

a(1) = 1, a(2) = 0, a(3) = 1; a(n) = (1/(n - 3)) * Sum_{k=1..n-3} ( Sum_{d|k} d * a(d) ) * a(n-k).

a(n) ~ c * d^n / n^(3/2), where d = 1.82975393308934955558864748939303527364978309460948926333116466766295641... and c = 0.8335864368398390652263577663136791087027831725508605623969711758177... - Vaclav Kotesovec, Jul 06 2021

Maple

a:= proc(n) option remember; `if`(n<4, [1, 0, 1][n], add(a(n-k)*
       add(d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
    end:
seq(a(n), n=1..42);  # Alois P. Heinz, Jul 01 2021

Mathematica

nmax = 42; A[_] = 0; Do[A[x_] = x + x^3 Exp[Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[2] = 0; a[3] = 1; a[n_] := a[n] = (1/(n - 3)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 42}]

Crossrefs

Cf. A007562, A218020, A346030.

Sequence in context: A050193 A173173 A303025 * A192669 A339484 A072164

Adjacent sequences: A346017 A346018 A346019 * A346021 A346022 A346023

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 01 2021

Status

approved