login
A325616
Triangle read by rows where T(n,k) is the number of length-k integer partitions of n into factorial numbers.
10
1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, 2, 1
OFFSET
0,61
FORMULA
T(n,k) is the coefficient of x^n * y^k in the expansion of Product_{i > 0} 1/(1 - y * x^(i!)).
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 0 1 1
0 0 1 1 1
0 0 0 1 1 1
0 1 0 1 1 1 1
0 0 1 0 1 1 1 1
0 0 1 1 1 1 1 1 1
0 0 0 1 1 1 1 1 1 1
0 0 0 1 1 2 1 1 1 1 1
0 0 0 0 1 1 2 1 1 1 1 1
0 0 1 0 1 1 2 2 1 1 1 1 1
0 0 0 1 0 1 1 2 2 1 1 1 1 1
0 0 0 1 1 1 1 2 2 2 1 1 1 1 1
0 0 0 0 1 1 1 1 2 2 2 1 1 1 1 1
0 0 0 0 1 1 2 1 2 2 2 2 1 1 1 1 1
0 0 0 0 0 1 1 2 1 2 2 2 2 1 1 1 1 1
0 0 0 1 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1
0 0 0 0 1 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1
0 0 0 0 1 1 1 1 2 2 3 2 2 2 2 2 1 1 1 1 1
Row n = 12 counts the following partitions:
(66)
(6222)
(62211)
(222222) (621111)
(2222211) (6111111)
(22221111)
(222111111)
(2211111111)
(21111111111)
(111111111111)
MATHEMATICA
Table[SeriesCoefficient[Product[1/(1-y*x^(i!)), {i, 1, n}], {x, 0, n}, {y, 0, k}], {n, 0, 15}, {k, 0, n}]
CROSSREFS
Row sums are A064986.
Cf. A008284.
Reciprocal factorial sum: A325618, A325619, A325620, A325622.
Sequence in context: A352516 A086075 A316865 * A056978 A321761 A037819
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, May 12 2019
STATUS
approved