A157394 A partition product of Stirling_1 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

1, 1, 4, 1, 12, 12, 1, 72, 48, 24, 1, 280, 600, 120, 24, 1, 1740, 4560, 1800, 144, 0, 1, 8484, 40740, 21000, 2520, 0, 0, 1, 57232, 390432, 223440, 33600, 0, 0, 0, 1, 328752, 3811248, 2845584, 438480, 0, 0, 0, 0, 1, 2389140

Offset: 1

Author

Peter Luschny, Mar 07 2009, Mar 14 2009

Keywords

• easy
• nonn
• tabl

Comments

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 4,

summed over parts with equal biggest part (see the Luschny link).

Underlying partition triangle is A144878.

Same partition product with length statistic is A049424.

Diagonal a(A000217(n)) = falling_factorial(4,n-1), row in A008279

Row sum is A049427.

Examples

			1
1       4
1      12       12
1      72       48       24
1     280      600      120      24
1    1740     4560     1800     144  0
1    8484    40740    21000    2520  0  0
1   57232   390432   223440   33600  0  0  0
1  328752  3811248  2845584  438480  0  0  0  0
1  2389140
		

Links

• Peter Luschny, Counting with Partitions.
• Peter Luschny, Generalized Stirling_1 Triangles.

Crossrefs

Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395

Formula

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n

T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that

1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),

f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n+6).