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A157400 A partition product with biggest-part statistic of Stirling_1 type (with parameter k = -2) as well as of Stirling_2 type (with parameter k = -2), (triangle read by rows). 25
1, 1, 2, 1, 6, 6, 1, 24, 24, 24, 1, 80, 180, 120, 120, 1, 330, 1200, 1080, 720, 720, 1, 1302, 7770, 10920, 7560, 5040, 5040, 1, 5936, 57456, 102480, 87360, 60480, 40320, 40320, 1, 26784, 438984, 970704, 1103760, 786240, 544320, 362880, 362880 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Partition product of prod_{j=0..n-1}((k+1)*j - 1) and n! at k = -2, summed over parts with equal biggest part (Stirling_2 type) as well as partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -2 (Stirling_1 type).

It shares this property with the signless Lah numbers.

Underlying partition triangle is A130561.

Same partition product with length statistic is A105278.

Diagonal a(A000217) = A000142.

Row sum is A000262.

LINKS

Table of n, a(n) for n=1..45.

Peter Luschny, Counting with Partitions.

Peter Luschny, Generalized Stirling_1 Triangles.

Peter Luschny, Generalized Stirling_2 Triangles.

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}(-j-1)

OR f_n = product_{j=0..n-2}(j-n) since both have the same absolute value n!.

E.g.f. of column k: exp((x^(k+1)-x)/(x-1))-exp((x^k-x)/(x-1)). - Alois P. Heinz, Oct 10 2015

EXAMPLE

Triangle starts:

1;

1, 2;

1, 6, 6;

1, 24, 24, 24;

1, 80, 180, 120, 120;

1, 330, 1200, 1080, 720, 720;

...

MAPLE

egf:= k-> exp((x^(k+1)-x)/(x-1))-exp((x^k-x)/(x-1)):

T:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):

seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Oct 10 2015

MATHEMATICA

egf[k_] := Exp[(x^(k+1)-x)/(x-1)] - Exp[(x^k-x)/(x-1)]; T[n_, k_] := n! * SeriesCoefficient[egf[k], {x, 0, n}]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-Fran├žois Alcover, Oct 11 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A157396, A157397, A157398, A157399, A080510, A157401, A157402, A157403, A157404, A157405, A157386, A157385, A157384, A157383, A126074, A157391, A157392, A157393, A157394, A157395.

Sequence in context: A110098 A244888 A130561 * A091599 A048999 A066667

Adjacent sequences:  A157397 A157398 A157399 * A157401 A157402 A157403

KEYWORD

easy,nonn,tabl

AUTHOR

Peter Luschny, Mar 09 2009, Mar 14 2009

STATUS

approved

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Last modified September 20 16:14 EDT 2017. Contains 292276 sequences.