A167137 - OEIS

%I #31 Jul 27 2024 13:55:48

%S 1,1,5,31,257,2551,30065,407191,6214577,105530071,1972879025,

%T 40213910551,886979957297,21044674731991,534313527291185,

%U 14448883517785111,414475305054698417,12568507978358276311,401658204472560090545,13490011548122407566871,474964861088609044357937,17491333169997896126211031

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

%C 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.

%C From _Peter Bala_, Jul 07 2022: (Start)

%C 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).

%C 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)

%H Vaclav Kotesovec, <a href="/A167137/b167137.txt">Table of n, a(n) for n = 0..400</a>

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

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

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

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

%F From _Peter Bala_, Sep 18 2013: (Start)

%F 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.

%F 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)

%F From _Vaclav Kotesovec_, Jun 17 2018: (Start)

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

%F 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)

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

%e A(log(1+x)) = P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 +...

%t Table[Sum[PartitionsP[k]*StirlingS2[n, k]*k!, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, May 21 2018 *)

%t nmax = 20; CoefficientList[Series[1/QPochhammer[E^x - 1], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Nov 22 2021 *)

%o (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))}

%o (PARI) {Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}

%o {a(n)=sum(k=0,n,numbpart(k)*Stirling2(n, k)*k!)}

%o (PARI) x='x+O('x^66); Vec( serlaplace( 1/eta(exp(x)-1) ) ) \\ _Joerg Arndt_, Sep 18 2013

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 03 2009