A090215 A generalization of triangles A071951 (Legendre-Stirling) and A089504.

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, 9702226427904, 1026160275456, 11956045824, 24697728, 11088, 1

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..36.

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.

Wolfdieter 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

Extensions

More terms coming from a-file added by Michel Marcus, Feb 08 2023

Status

approved