A294394 -id:A294394 - cp's OEIS frontend

A294392 E.g.f.: exp(Sum_{n>=1} A001227(n) * x^n).

1, 1, 3, 19, 97, 801, 7411, 73123, 821409, 10977697, 151612291, 2286137811, 38308830913, 669163118209, 12649211055027, 257559356068771, 5432325991339201, 121949878889492673, 2907330680764076419, 71860237654425159187, 1871308081194213959841

Offset: 0

Seiichi Manyama, Oct 30 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..441

E.g.f.: exp(Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): this sequence (k=0), A294394 (k=1), A294395 (k=2).

Cf. A001227, A206303.

a[n_] := a[n] = If[n == 0, 1, Sum[k*DivisorSum[k, Mod[#, 2] &]*a[n - k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2018 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, d%2)*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A001227(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(x^(2*k-1)/(1 - x^(2*k-1))). - Ilya Gutkovskiy, Nov 27 2017
Conjecture: log(a(n)/n!) ~ sqrt(n*log(n)). - Vaclav Kotesovec, Sep 07 2018

A294395 E.g.f.: exp(Sum_{n>=1} A050999(n) * x^n).

Original entry on oeis.org

1, 1, 3, 67, 289, 5121, 71731, 861043, 18134817, 303946849, 6724342531, 146426154051, 3533373668353, 93259190078497, 2489644674735219, 75193364720030131, 2265438714279130561, 74716734198386887233, 2543592184722884351107, 90853513680763023292099

Offset: 0

Seiichi Manyama, Oct 30 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..422

E.g.f.: exp(Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): A294392 (k=0), A294394 (k=1), this sequence (k=2).

Cf. A050999, A262811.

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, d^2*(d%2))*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A050999(k)*a(n-k)/(n-k)! for n > 0.

a(n) ~ (3*zeta(3))^(1/8) * n^(n - 1/8) / (2*exp(n - 4*zeta(3)^(1/4) * n^(3/4) / 3^(3/4) - n^(1/4) / (4*3^(5/4)*zeta(3)^(1/4)))). - Vaclav Kotesovec, Nov 01 2024

A294460 E.g.f.: exp(-Sum_{n>=1} A000593(n) * x^n).

Original entry on oeis.org

1, -1, -1, -19, 73, -401, 5191, -29779, 879089, -7232833, 103048111, -1891058291, 31696845049, -649348332049, 9310670445623, -270217657103731, 5480877008565601, -131578355696804609, 3133521575795986399, -81890613282163881043, 2460096066325021029161

Offset: 0

Seiichi Manyama, Oct 31 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..437

E.g.f.: exp(-Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): A294459 (k=0), this sequence (k=1), A294461 (k=2).

Cf. A000593, A081362, A294394.

N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, d*(d%2))*x^k))))

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A000593(k)*a(n-k)/(n-k)! for n > 0.

cp's OEIS Frontend

A294392 E.g.f.: exp(Sum_{n>=1} A001227(n) * x^n).

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A294395 E.g.f.: exp(Sum_{n>=1} A050999(n) * x^n).

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A294460 E.g.f.: exp(-Sum_{n>=1} A000593(n) * x^n).

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