A157397 A partition product of Stirling_2 type [parameter k = -5] with biggest-part statistic (triangle read by rows).

1, 1, 5, 1, 15, 45, 1, 105, 180, 585, 1, 425, 2700, 2925, 9945, 1, 3075, 34650, 52650, 59670, 208845, 1, 15855, 308700, 1248975, 1253070, 1461915, 5221125, 1, 123515, 4475520, 23689575, 33972120, 35085960, 41769000

Offset: 1

Author

Peter Luschny, Mar 09 2009

Keywords

• easy
• nonn
• tabl

Comments

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -5,

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

Underlying partition triangle is A134273.

Same partition product with length statistic is A049029.

Diagonal a(A000217) = A007696.

Row sum is A049120.

Links

• Peter Luschny, Counting with Partitions.
• Peter Luschny, Generalized Stirling_2 Triangles.

Crossrefs

Cf. A157396, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157404, A157405

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-1}(-4*j - 1).

Extensions

Offset corrected by Peter Luschny, Mar 14 2009