A089504 A generalization of triangle A071951 (Legendre-Stirling).

1, 6, 1, 36, 30, 1, 216, 756, 90, 1, 1296, 18360, 6156, 210, 1, 7776, 441936, 387720, 31356, 420, 1, 46656, 10614240, 23705136, 4150440, 119556, 756, 1, 279936, 254788416, 1432922400, 521757936, 29257200, 373572, 1260, 1, 1679616

Offset

1,2

Comments

This triangle underlies the array entry A078741 ((3,3)-generalized Stirling2).

For the computation of the column sequences see A089505.

Links

Table of n, a(n) for n=1..37.

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_{r=1..m} 1-fallfac(r+2, 3)*x with fallfac(n, k) := A008279(n, k) (falling factorials).

a(n, m) = Sum_{p=1..m} A089505(m, p)*((p+2)*(p+1)*p)^(n-m))/D(m) if n>=m>=1 else 0; with D(m) := A089506(m).

Example

[1]; [6,1]; [36,30,1]; [216,756,90,1]; ...
a(3,2) = 30 = ((-1)*(3*2*1)^1 + 4*(4*3*2)^1)/3.

Mathematica

max = 10; f[m_] := 1/Product[1 - FactorialPower[r + 2, 3]*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).

The column sequences (without leading zeros) are A000400 (powers of 6), A089507, A089513-4, etc.

Cf. A008279, A089505, A089506.

Sequence in context: A147320 A038255 A075501 * A145927 A113365 A293172

Adjacent sequences: A089501 A089502 A089503 * A089505 A089506 A089507

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Dec 01 2003

Status

approved