A167137 E.g.f.: P(exp(x)-1) where P(x) is the g.f. of the partition numbers (A000041).

1, 1, 5, 31, 257, 2551, 30065, 407191, 6214577, 105530071, 1972879025, 40213910551, 886979957297, 21044674731991, 534313527291185, 14448883517785111, 414475305054698417, 12568507978358276311, 401658204472560090545, 13490011548122407566871, 474964861088609044357937, 17491333169997896126211031

Offset

0,3

Comments

CONJECTURE: Sum_{n>=0} a(n)^m * log(1+x)^n/n! is an integer series in x for all integer m>0; see A167138 and A167139 for examples.

From Peter Bala, Jul 07 2022: (Start)

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 16 we obtain the sequence [1, 1, 5, 15, 1, 7, 1, 7, 1, 7, ...], with an apparent period of 2 beginning at a(4).

More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series. (End)

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Formula

a(n) = Sum_{k=0..n} A000041(k)*Stirling2(n,k)*k! where A000041 is the partition numbers.

E.g.f.: exp( Sum_{n>=1} sigma(n)*[exp(x)-1]^n/n ).

Sum_{n>=0} a(n) * log(1+x)^n/n! = g.f. of the partition numbers (A000041).

Sum_{n>=0} a(n)^2*log(1+x)^n/n! = g.f. of A167138.

From Peter Bala, Sep 18 2013: (Start)

Sum {n >= 0} (-1)^n*a(n)*(log(1 - x))^n/n! = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + ... is the o.g.f. of A218482.

a(n) is always odd. Congruences for n >= 1: a(2*n) = 2 (mod 3); a(4*n) = 2 (mod 5); a(6*n) = 0 (mod 7); a(10*n) = 7 (mod 11); a(12*n) = 5 (mod 13); a(16*n) = 0 (mod 17). (End)

From Vaclav Kotesovec, Jun 17 2018: (Start)

a(n) ~ n! * exp((1/log(2) - 1) * Pi^2 / 24 + Pi*sqrt(n/(3*log(2)))) / (4 * sqrt(3) * n * (log(2))^n).

a(n) ~ sqrt(Pi) * exp((1/log(2) - 1) * Pi^2 / 24 + Pi*sqrt(n/(3*log(2))) - n) * n^(n + 1/2) / (2^(3/2) * sqrt(3) * n * (log(2))^n). (End)

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 31*x^3/3! + 257*x^4/4! +...
A(log(1+x)) = P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 +...

Mathematica

Table[Sum[PartitionsP[k]*StirlingS2[n, k]*k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 21 2018 *)
nmax = 20; CoefficientList[Series[1/QPochhammer[E^x - 1], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 22 2021 *)

Prog

(PARI) {a(n)=if(n==0, 1, n!*polcoeff(exp(sum(m=1, n, sigma(m)*(exp(x+x*O(x^n))-1)^m/m) ), n))}
(PARI) {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, numbpart(k)*Stirling2(n, k)*k!)}
(PARI) x='x+O('x^66); Vec( serlaplace( 1/eta(exp(x)-1) ) ) \\ Joerg Arndt, Sep 18 2013

Crossrefs

Cf. A167138, A000041, A019538 (Stirling2), A218482, A305550, A306045, A320349.

Sequence in context: A056541 A291885 A126121 * A279434 A000556 A320512

Adjacent sequences: A167134 A167135 A167136 * A167138 A167139 A167140

Keyword

nonn

Author

Paul D. Hanna, Nov 03 2009

Status

approved