A300514 -id:A300514 - cp's OEIS frontend

A300515 Expansion of e.g.f. log(Sum_{k>=0} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).

0, 1, 0, 1, -3, 7, -24, 130, -748, 4446, -30694, 245586, -2131621, 19850237, -201363613, 2214638141, -26037523804, 325653856386, -4331545709166, 61069238694738, -908488414975896, 14220161323121232, -233746798117055047, 4025924893291859919, -72487584601341680720

Offset: 0

Ilya Gutkovskiy, Mar 07 2018

sign

Logarithmic transform of A000009.

			E.g.f.: A(x) = x/1! + x^3/3! - 3*x^4/4! + 7*x^5/5! - 24*x^6/6! + 130*x^7/7! - 748*x^8/8! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..470
N. J. A. Sloane, Transforms
Index entries for related partition-counting sequences

Cf. A000009, A265024, A300512, A300514.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(
      j*a(j)*binomial(n, j)*t(n-j), j=1..n-1)/n))(b)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 07 2018

nmax = 24; CoefficientList[Series[Log[Sum[PartitionsQ[k] x^k/k!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = PartitionsQ[n] - Sum[k Binomial[n, k] PartitionsQ[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 24}]

E.g.f.: log(Sum_{k>=0} A000009(k)*x^k/k!).

A324237 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, 1, 3, 14, 84, 633, 5730, 60485, 729710, 9904064, 149358998, 2477662364, 44837516675, 879028693860, 18558771941586, 419815668642109, 10129704474860688, 259695154154923814, 7049438079064414206, 201988316828399901634, 6092203404985463075656

Offset: 0

Ilya Gutkovskiy, Sep 02 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..419

Cf. A000009, A266232, A300514, A304969, A324236.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n, k)*b(k), k=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 02 2019

nmax = 21; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * q(k) * a(n-k).

a(n) ~ c * d^n * n!, where d = 1.5080583621492799630678624980320180394686208919872154400104169910221003637... and c = 0.67652958824662835367141799671720225317465169475061770258661897351... - Vaclav Kotesovec, Sep 03 2019

cp's OEIS Frontend

A300515 Expansion of e.g.f. log(Sum_{k>=0} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).

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