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

Author

Paul D. Hanna, Nov 03 2009

Keywords

• nonn

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)

Examples

			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 +...
		

Links

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

Crossrefs

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

Programs

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 *)

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

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)