A157391 A partition product of Stirling_1 type [parameter k = 1] with biggest-part statistic (triangle read by rows).

1, 1, 1, 1, 3, 0, 1, 9, 0, 0, 1, 25, 0, 0, 0, 1, 75, 0, 0, 0, 0, 1, 231, 0, 0, 0, 0, 0, 1, 763, 0, 0, 0, 0, 0, 0, 1, 2619, 0, 0, 0, 0, 0, 0, 0, 1, 9495, 0, 0, 0, 0, 0, 0, 0, 0, 1, 35695, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 140151

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 = 1,

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

Underlying partition triangle is A144357.

Same partition product with length statistic is A049403.

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

Row sum is A000085.

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+3).