login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A167137 E.g.f.: P(exp(x)-1) where P(x) is the g.f. of the partition numbers (A000041). 14
1, 1, 5, 31, 257, 2551, 30065, 407191, 6214577, 105530071, 1972879025, 40213910551, 886979957297, 21044674731991, 534313527291185, 14448883517785111, 414475305054698417, 12568507978358276311, 401658204472560090545, 13490011548122407566871, 474964861088609044357937, 17491333169997896126211031 (list; graph; refs; listen; history; text; internal format)
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.

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 *)

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 28 03:34 EST 2020. Contains 331315 sequences. (Running on oeis4.)