|
| |
|
|
A122399
|
|
a(n) = Sum_{k=0..n} k^n * k! * Stirling2(n,k).
|
|
34
|
|
|
|
1, 1, 9, 211, 9285, 658171, 68504709, 9837380491, 1863598406805, 450247033371451, 135111441590583909, 49300373690091496171, 21495577955682021043125, 11037123350952586270549531, 6591700149366720366704735109
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 2, 1, 3, 3, 0, 1, 2, 1, 3, 3, 0, ...], with an apparent period of 6. Cf. A338040. - Peter Bala, May 31 2022
|
|
|
LINKS
|
|
|
|
FORMULA
|
E.g.f.: Sum_{n>=0} exp(n^2*x) / (1 + exp(n*x))^(n+1). - Paul D. Hanna, Oct 26 2014
E.g.f.: Sum_{n>=0} exp(-n*x) / (1 + exp(-n*x))^(n+1). - Paul D. Hanna, Oct 30 2014
O.g.f.: Sum_{n>=0} n^n * n! * x^n / Product_{k=1..n} (1 - n*k*x). - Paul D. Hanna, Jan 05 2013
Limit n->infinity (a(n)/n!)^(1/n)/n = ((1+exp(1/r))*r^2)/exp(1) = A317855/exp(1) = 1.162899527477400818845..., where r = 0.87370243323966833... is the root of the equation 1/(1+exp(-1/r)) = -r*LambertW(-exp(-1/r)/r). - Vaclav Kotesovec, Jun 21 2013
a(n) ~ c * A317855^n * (n!)^2 / sqrt(n), where c = 0.327628285569869481442286492410507030710253054522608... - Vaclav Kotesovec, Aug 09 2018
Let A(x) = 1 + x + 9*x^2/2! + 211*x^3/3! + ... denote the e.g.f. of the sequence. Let F(x) denote the series reversion of A(x) - 1 = x - 9*x^2/2 + 16*x^3/3 - 205*x^4/4 - 2714*x^5/5 - .... Then both dF/dx = 1 - 9*x + 16*x^2 - 205*x^3 - 2714*x^4 - ... and exp(F(x)) = 1 + x - 4*x^2 + x^3 - 38*x^4 - 606*x^5 - ... have integer coefficients. Note that 1 + series reversion(exp(F(x)) - 1) is the o.g.f. for A122400. - Peter Bala, Aug 09 2022
|
|
|
EXAMPLE
|
E.g.f.: A(x) = 1 + x + 9*x^2/2! + 211*x^3/3! + 9285*x^4/4! + 658171*x^5/5! + ...
such that
A(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2 + (exp(3*x)-1)^3 + (exp(4*x)-1)^4 + ...
The e.g.f. is also given by the series:
A(x) = 1/2 + exp(x)/(1+exp(x))^2 + exp(4*x)/(1+exp(2*x))^3 + exp(9*x)/(1+exp(3*x))^4 + exp(16*x)/(1+exp(4*x))^5 + exp(25*x)/(1+exp(5*x))^6 + ...
or, equivalently,
A(x) = 1/2 + exp(-x)/(1+exp(-x))^2 + exp(-2*x)/(1+exp(-2*x))^3 + exp(-3*x)/(1+exp(-3*x))^4 + exp(-4*x)/(1+exp(-4*x))^5 + exp(-5*x)/(1+exp(-5*x))^6 + ...
|
|
|
MAPLE
|
a := n -> add(k^n*k!*combinat[stirling2](n, k), k=0..n); # Max Alekseyev, Feb 01 2007
|
|
|
MATHEMATICA
|
Flatten[{1, Table[Sum[k^n*k!*StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jun 21 2013 *)
|
|
|
PROG
|
(PARI) {a(n)=polcoeff(sum(m=0, n, m^m*m!*x^m/prod(k=1, m, 1-m*k*x+x*O(x^n))), n)}
(PARI) {a(n)=n!*polcoeff(sum(k=0, n, (exp(k*x +x*O(x^n)) - 1)^k), n)}
(PARI) /* From e.g.f. infinite series: */
\p100 \\ set precision
{A=Vec(serlaplace(sum(n=0, 500, 1.*exp(n^2*x +O(x^26))/(1 + exp(n*x +O(x^26)))^(n+1)) ))}
for(n=0, #A-1, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 30 2014
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
