|
| |
|
|
A065143
|
|
a(n) = Sum_{k=0..n} Stirling2(n,k)*(1+(-1)^k)*2^k/2.
|
|
12
|
|
|
|
1, 0, 4, 12, 44, 220, 1228, 7196, 45004, 303900, 2201676, 16920860, 136966860, 1163989788, 10364408140, 96463232284, 935872773068, 9440653262620, 98809201693260, 1071131795708188, 12007932126074060
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Stirling transform of A199572 (aerated powers of 4).
|
|
|
LINKS
|
|
|
|
FORMULA
|
Representation as a sum of an infinite series: a(n) = exp(2)*Sum_{k = 0..infinity} ((2*k)^n*2^(2*k)/(2*k)!) - sinh(2)*sum_{k = 0..infinity}(k^n*2^k/k!), for n >= 0.
a(n) ~ n^n * cosh(2*exp(r)-2) / (r^n * (exp(n) * sqrt(4*exp(2*r)*r^2/n + 1-n+r))), where r is the root of the equation -2*exp(r)*r*tanh(2-2*exp(r)) = n.
(a(n)/n!)^(1/n) ~ exp(1/LambertW(n/2)) / LambertW(n/2).
(End)
a(n) = (Bell_n(2) + Bell_n(-2))/2, where Bell_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Nov 01 2015
|
|
|
MATHEMATICA
|
Table[Sum[StirlingS2[n, k]*(1+(-1)^k)*2^k/2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2014 *)
|
|
|
PROG
|
(PARI) a(n) = sum(k=0, n, stirling(n, k, 2)*(1+(-1)^k)*2^k/2); \\ Michel Marcus, Nov 02 2015
(PARI) x='x+O('x^50); Vec(serlaplace(cosh(2*exp(x)-2))) \\ G. C. Greubel, Nov 16 2017
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
