A325616 - OEIS

%I #6 May 13 2019 01:10:35

%S 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,

%T 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,

%U 0,0,1,1,2,1,1,1,1,1,0,0,1,0,1,1,2,2,1

%N Triangle read by rows where T(n,k) is the number of length-k integer partitions of n into factorial numbers.

%F T(n,k) is the coefficient of x^n * y^k in the expansion of Product_{i > 0} 1/(1 - y * x^(i!)).

%e Triangle begins:

%e   1

%e   0 1

%e   0 1 1

%e   0 0 1 1

%e   0 0 1 1 1

%e   0 0 0 1 1 1

%e   0 1 0 1 1 1 1

%e   0 0 1 0 1 1 1 1

%e   0 0 1 1 1 1 1 1 1

%e   0 0 0 1 1 1 1 1 1 1

%e   0 0 0 1 1 2 1 1 1 1 1

%e   0 0 0 0 1 1 2 1 1 1 1 1

%e   0 0 1 0 1 1 2 2 1 1 1 1 1

%e   0 0 0 1 0 1 1 2 2 1 1 1 1 1

%e   0 0 0 1 1 1 1 2 2 2 1 1 1 1 1

%e   0 0 0 0 1 1 1 1 2 2 2 1 1 1 1 1

%e   0 0 0 0 1 1 2 1 2 2 2 2 1 1 1 1 1

%e   0 0 0 0 0 1 1 2 1 2 2 2 2 1 1 1 1 1

%e   0 0 0 1 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1

%e   0 0 0 0 1 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1

%e   0 0 0 0 1 1 1 1 2 2 3 2 2 2 2 2 1 1 1 1 1

%e Row n = 12 counts the following partitions:

%e   (66)

%e   (6222)

%e   (62211)

%e   (222222) (621111)

%e   (2222211) (6111111)

%e   (22221111)

%e   (222111111)

%e   (2211111111)

%e   (21111111111)

%e   (111111111111)

%t Table[SeriesCoefficient[Product[1/(1-y*x^(i!)),{i,1,n}],{x,0,n},{y,0,k}],{n,0,15},{k,0,n}]

%Y Row sums are A064986.

%Y Cf. A008284.

%Y Factorial numbers: A000142, A007489, A076934, A108731, A115944, A227157, A284605, A322583, A325509, A325617.

%Y Reciprocal factorial sum: A325618, A325619, A325620, A325622.

%K nonn,tabl

%O 0,61

%A _Gus Wiseman_, May 12 2019