|
| |
|
|
A259401
|
|
a(n) = Sum_{k=0..n} 2^(n-k)*p(k), where p(k) is the partition function A000041.
|
|
4
|
|
|
|
1, 3, 8, 19, 43, 93, 197, 409, 840, 1710, 3462, 6980, 14037, 28175, 56485, 113146, 226523, 453343, 907071, 1814632, 3629891, 7260574, 14522150, 29045555, 58092685, 116187328, 232377092, 464757194, 929518106, 1859040777, 3718087158, 7436181158, 14872370665
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
In general, Sum_{k=0..n} (m^(n-k) * p(k)) ~ m^n / QPochhammer[1/m, 1/m], for m > 1.
|
|
|
LINKS
|
Alois P. Heinz, Table of n, a(n) for n = 0..3320
|
|
|
FORMULA
|
a(n) ~ c * 2^n, where c = 1/A048651 = 1/QPochhammer[1/2, 1/2] = 3.462746619455...
G.f.: (1/(1 - 2*x)) * Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Dec 03 2019
|
|
|
MAPLE
|
a:= proc(n) option remember; `if`(n<0, 0,
2*a(n-1)+combinat[numbpart](n))
end:
seq(a(n), n=0..32); # Alois P. Heinz, Dec 03 2019
|
|
|
MATHEMATICA
|
Table[Sum[2^(n-k)*PartitionsP[k], {k, 0, n}], {n, 0, 50}]
|
|
|
PROG
|
(PARI) a(n) = sum(k=0, n, 2^(n-k)*numbpart(k)); \\ Michel Marcus, Dec 03 2019
|
|
|
CROSSREFS
|
Cf. A000041, A048651, A090764, A259400, A292746.
Sequence in context: A065352 A161993 A360489 * A008466 A102712 A054480
Adjacent sequences: A259398 A259399 A259400 * A259402 A259403 A259404
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Vaclav Kotesovec, Jun 26 2015
|
|
|
STATUS
|
approved
|
| |
|
|
