A346031 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, 0, 1, 2, 1, 1, 4, 4, 3, 9, 14, 12, 22, 43, 49, 66, 130, 186, 234, 406, 663, 884, 1362, 2303, 3347, 4884, 8049, 12478, 18240, 28853, 46075, 69163, 106470, 170305, 262853, 401773, 635780, 998609, 1536093, 2405345, 3801601, 5910267, 9212253, 14548179, 22818301

Offset

1,7

Links

Table of n, a(n) for n=1..48.

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} (-1)^(k/d+1) * d * a(d) ) * a(n-k).

Maple

a:= proc(n) option remember; `if`(n<4, [1, 0, 1][n], add(a(n-k)*add(
     (-1)^(k/d+1)*d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
    end:
seq(a(n), n=1..48);  # Alois P. Heinz, Jul 01 2021

Mathematica

nmax = 48; A[_] = 0; Do[A[x_] = x + x^3 Exp[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] = 1; a[2] = 0; a[3] = 1; a[n_] := a[n] = (1/(n - 3)) Sum[Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 48}]

Crossrefs

Cf. A007560, A316075, A346032.

Sequence in context: A294217 A123246 A122518 * A129704 A144460 A336996

Adjacent sequences: A346028 A346029 A346030 * A346032 A346033 A346034

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 01 2021

Status

approved