|
|
A267709
|
|
Number of partitions of pentagonal numbers.
|
|
1
|
|
|
1, 1, 7, 77, 1002, 14883, 239943, 4087968, 72533807, 1327710076, 24908858009, 476715857290, 9275102575355, 182973889854026, 3652430836071053, 73653287861850339, 1498478743590581081, 30724985147095051099, 634350763653787028583, 13177726323474524612308
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
FORMULA
|
a(n) ~ exp((Pi*sqrt(n*(3*n - 1)))/sqrt(3))/(2*sqrt(3)*n*(3*n - 1)).
a(n) = [x^(n*(3*n-1)/2)] Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 11 2017
|
|
EXAMPLE
|
a(2) = 7, because second pentagonal number is a 5 and 5 can be partitioned in 7 distinct ways: 5, 4 + 1, 3 + 2, 3 + 1 + 1, 3 + 2 + 1, 2 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1.
|
|
MATHEMATICA
|
Table[PartitionsP[n ((3 n - 1)/2)], {n, 0, 19}]
|
|
PROG
|
(Python)
from sympy.ntheory import npartitions
print([npartitions(n*(3*n - 1)//2) for n in range(51)]) # Indranil Ghosh, Apr 11 2017
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|