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A061692 Triangle of generalized Stirling numbers. 4
1, 1, 4, 1, 27, 36, 1, 172, 864, 576, 1, 1125, 17500, 36000, 14400, 1, 7591, 351000, 1746000, 1944000, 518400, 1, 52479, 7197169, 80262000, 191394000, 133358400, 25401600, 1, 369580, 151633440, 3691514176, 17188416000, 23866214400, 11379916800, 1625702400 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Alois P. Heinz, Rows n = 1..100, flattened

J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

FORMULA

T(n, k) = 1/k!*Sum multinomial(n, n_1, n_2, ..n_k)^3, where the sum extends over all compositions (n_1, n_2, .., n_k) of n into exactly k nonnegative parts. - Vladeta Jovovic, Apr 23 2003

The row polynomials R(n,x) satisfy the recurrence equation R(n,x) = x*( sum {k = 0..n-1} binomial(n,k)^2*binomial(n-1,k)*R(k,x) ) with R(0,x) = 1. Also R(n,x + y) = sum {k = 0..n} binomial(n,k)^3*R(k,x)*R(n-k,y). - Peter Bala, Sep 17 2013

EXAMPLE

1; 1,4; 1,27,36; 1,172,864,576; ...

MAPLE

b:= proc(n) option remember; expand(

      `if`(n=0, 1, add(x*b(n-i)/i!^3, i=1..n)))

    end:

T:= n-> (p-> seq(coeff(p, x, i)/i!, i=1..n))(b(n)*n!^3):

seq(T(n), n=1..10);  # Alois P. Heinz, Sep 10 2019

MATHEMATICA

R[0, _] = 1; R[n_, x_] := R[n, x] = x*Sum[Binomial[n, k]^2*Binomial[n-1, k]*R[k, x], {k, 0, n-1}]; Table[CoefficientList[R[n, x], x] // Rest, {n, 1, 8}] // Flatten (* Jean-Fran├žois Alcover, Sep 01 2015, after Peter Bala *)

CROSSREFS

Diagonals give A001044, A061695, A061693, A061694. Cf. A061691.

Row sums give A061684.

Sequence in context: A225213 A137906 A139051 * A096206 A134150 A134151

Adjacent sequences:  A061689 A061690 A061691 * A061693 A061694 A061695

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jun 19 2001

EXTENSIONS

More terms from Vladeta Jovovic, Apr 23 2003

STATUS

approved

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Last modified July 12 10:09 EDT 2020. Contains 335657 sequences. (Running on oeis4.)