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A090215 A generalization of triangles A071951 (Legendre-Stirling) and A089504. 5
1, 24, 1, 576, 144, 1, 13824, 17856, 504, 1, 331776, 2156544, 199296, 1344, 1, 7962624, 259117056, 73903104, 1328256, 3024, 1, 191102976, 31102009344, 26864234496, 1189638144, 6408576, 6048, 1, 4586471424, 3732432224256 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This triangle underlies the array entry A090214 ((4,4)-generalized Stirling2).

LINKS

Table of n, a(n) for n=1..30.

R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv preprint arXiv:1302.4694 [math.CO], 2013.

R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.

W. Lang, First 8 rows.

FORMULA

G.f. for m-th column sequence (without leading zeros and m>=1) is 1/product(1-fallfac(r+3, 4)*x, r=1..m) with fallfac(n, k) := A008279(n, k) (falling factorials).

a(n, m) = sum(A089515(m, p)*fallfac(p, 4)^(n-m), p=1..m)/D(m) if n>=m>=1 else 0; with D(m) := A089516(m).

EXAMPLE

[1]; [24,1]; [576,144,1]; [13824,17856,504,1]; ...

MATHEMATICA

max = 10; f[m_] := 1/Product[1-FactorialPower[r+3, 4]*x, {r, 1, m}]; col[m_] := CoefficientList[f[m] + O[x]^(max-m+1), x]; a[n_, m_] := col[m][[n-m+1]]; Table[a[n, m], {n, 1, max}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

CROSSREFS

Cf. A071951 (Legendre-Stirling, (2, 2) case), A089504 ((3, 3)-case).

The column sequences (without leading zeros) are A009968 (powers of 24), etc.

Sequence in context: A076721 A232988 A292781 * A318105 A040570 A040569

Adjacent sequences:  A090212 A090213 A090214 * A090216 A090217 A090218

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Dec 01 2003

STATUS

approved

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Last modified December 6 19:27 EST 2021. Contains 349567 sequences. (Running on oeis4.)