A307817 a(1) = 1; a(n+1) = Sum_{d|n} sigma(n/d)*a(d), where sigma = sum of divisors (A000203).

1, 1, 4, 8, 18, 24, 52, 60, 106, 135, 213, 225, 397, 411, 599, 719, 1001, 1019, 1533, 1553, 2192, 2464, 3151, 3175, 4502, 4641, 5888, 6404, 8145, 8175, 11040, 11072, 13863, 14811, 17886, 18390, 23723, 23761, 28440, 30140, 36650, 36692, 45952, 45996, 55095, 58535, 68084, 68132, 83720, 84193

Offset

1,3

Links

Table of n, a(n) for n=1..50.

Formula

G.f. A(x) satisfies: A(x) = x * (1 + Sum_{k>=1} sigma(k)*A(x^k)).

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{i>=1} Sum_{j>=1} sigma(i)*a(j)*x^(i*j)).

Mathematica

a[n_] := a[n] = Sum[DivisorSigma[1, (n - 1)/d] a[d] , {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 50}]
terms = 50; A[_] = 0; Do[A[x_] = x (1 + Sum[DivisorSigma[1, k]  A[x^k], {k, 1, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[Sum[DivisorSigma[1, i] a[j] x^(i j), {j, 1, n - 1}], {i, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 50}]

Prog

(PARI) lista(nn) = { my(va=vector(nn)); va[1] = 1; for (n=2, nn, va[n] = sumdiv(n-1, d, sigma((n-1)/d)*va[d])); va; } \\ Michel Marcus, Apr 30 2019

Crossrefs

Cf. A000203, A007557, A307794.

Sequence in context: A308150 A070213 A077474 * A210433 A182099 A318756

Adjacent sequences: A307814 A307815 A307816 * A307818 A307819 A307820

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 30 2019

Status

approved