A157393 A partition product of Stirling_1 type [parameter k = 3] with biggest-part statistic (triangle read by rows).

1, 1, 3, 1, 9, 6, 1, 45, 24, 6, 1, 165, 240, 30, 0, 1, 855, 1560, 360, 0, 0, 1, 3843, 12180, 3360, 0, 0, 0, 1, 21819, 96096, 30660, 0, 0, 0, 0, 1, 114075, 794304, 318276, 0, 0, 0, 0, 0, 1, 703215, 6850080, 3270960, 0, 0, 0, 0, 0, 0, 1, 4125495, 62516520, 35053920, 0, 0

Offset: 1

Author

Peter Luschny, Mar 07 2009, Mar 14 2009

Keywords

• easy
• nonn
• tabl

Comments

Partition product prod_{j=0..n-2}(k-n+j+2) and n! at k = 3, summed over parts with equal biggest part (see the Luschny link).

Underlying partition triangle is A144877.

Same partition product with length statistic is A049410.

Diagonal a(A000217(n)) = falling_factorial(3,n-1), row in A008279.

Row sum is A049426.

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+5).