A308076 -id:A308076 - cp's OEIS frontend

A328424 a(1) = 1; a(n) = Sum_{d|n, d < n} p(n/d) * a(d), where p = A000041 (partition numbers).

1, 2, 3, 9, 7, 23, 15, 50, 39, 70, 56, 187, 101, 195, 218, 420, 297, 625, 490, 949, 882, 1226, 1255, 2533, 2007, 2840, 3217, 4588, 4565, 6966, 6842, 10099, 10479, 13498, 15093, 21507, 21637, 27975, 31791, 41722, 44583, 58022, 63261, 80415, 90799, 110578, 124754

Offset: 1

Ilya Gutkovskiy, Oct 15 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000041, A050354, A050369, A074206, A308076.

a[n_] := If[n == 1, n, Sum[If[d < n, PartitionsP[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 47}]
terms = 47; A[] = 0; Do[A[x] = x + Sum[PartitionsP[k] A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal,terms + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} p(k) * A(x^k).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). - Vaclav Kotesovec, Oct 16 2019

A351405 a(1) = 1; a(n+1) = Sum_{d|n} 2^(n/d - 1) * a(d).

Original entry on oeis.org

1, 1, 3, 7, 17, 33, 75, 139, 289, 557, 1119, 2143, 4341, 8437, 16843, 33343, 66573, 132109, 264243, 526387, 1052549, 2101617, 4202031, 8396335, 16792705, 33570193, 67137403, 134248191, 268492033, 536927489, 1073853307, 2147595131, 4295180241, 8590155085

Offset: 1

Ilya Gutkovskiy, Feb 10 2022

nonn

Cf. A003238, A038046, A054599, A308076.

a:= proc(n) option remember; `if`(n=1, 1,
      add(2^((n-1)/d-1)*a(d), d=numtheory[divisors](n-1)))
    end:
seq(a(n), n=1..34);  # Alois P. Heinz, Feb 10 2022

a[1] = 1; a[n_] := a[n] = Sum[2^((n - 1)/d - 1) a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 34}]
nmax = 34; A[] = 0; Do[A[x] = x (1 + Sum[2^(k - 1) A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x * ( 1 + A(x) + 2 * A(x^2) + 4 * A(x^3) + ... + 2^(k-1) * A(x^k) + ... ).
G.f.: x * ( 1 + Sum_{n>=1} a(n) * x^n / (1 - 2 * x^n) ).
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Feb 18 2022

A329803 a(1) = 1; a(n) = Sum_{d|n, d < n} q(n/d) * a(d), where q() = A000009.

Original entry on oeis.org

1, 1, 2, 3, 3, 8, 5, 11, 12, 16, 12, 37, 18, 32, 39, 55, 38, 90, 54, 105, 96, 113, 104, 236, 151, 201, 232, 301, 256, 450, 340, 517, 496, 588, 615, 988, 760, 972, 1054, 1395, 1260, 1766, 1610, 2078, 2240, 2512, 2590, 3653, 3289, 4029, 4249, 5038, 5120, 6526

Offset: 1

Ilya Gutkovskiy, Nov 21 2019

nonn

Cf. A000009, A050354, A050369, A074206, A308076, A328424.

a[n_] := If[n == 1, n, Sum[If[d < n, PartitionsQ[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 54}]
nmax = 54; A[] = 0; Do[A[x] = x + Sum[PartitionsQ[k] A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} q(k) * A(x^k).

cp's OEIS Frontend

A328424 a(1) = 1; a(n) = Sum_{d|n, d < n} p(n/d) * a(d), where p = A000041 (partition numbers).

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A351405 a(1) = 1; a(n+1) = Sum_{d|n} 2^(n/d - 1) * a(d).

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A329803 a(1) = 1; a(n) = Sum_{d|n, d < n} q(n/d) * a(d), where q() = A000009.

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