|
%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
|
