A048966 - OEIS

A048966

A convolution triangle of numbers obtained from A025748.

1, 3, 1, 15, 6, 1, 90, 39, 9, 1, 594, 270, 72, 12, 1, 4158, 1953, 567, 114, 15, 1, 30294, 14580, 4482, 1008, 165, 18, 1, 227205, 111456, 35721, 8667, 1620, 225, 21, 1, 1741905, 867834, 287199, 73656, 15075, 2430, 294, 24, 1, 13586859, 6857136, 2328183, 623106, 136323, 24354, 3465, 372, 27, 1

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OFFSET

1,2

COMMENTS

A generalization of the Catalan triangle A033184.

LINKS

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

FORMULA

a(n, m) = 3*(3*(n-1)-m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0; a(1, 1)=1.

G.f. for m-th column: ((1-(1-9*x)^(1/3))/3)^m.

a(n,m) = m/n * sum(k=0..n-m, binomial(k,n-m-k) * 3^k*(-1)^(n-m-k) * binomial(n+k-1,n-1)). - Vladimir Kruchinin, Feb 08 2011

EXAMPLE

Triangle begins:

3, 1;

15, 6, 1;

90, 39, 9, 1;

594, 270, 72, 12, 1;

4158, 1953, 567, 114, 15, 1;

MATHEMATICA

a[n_, m_] /; n >= m >= 1 := a[n, m] = 3*(3*(n-1) - m)*a[n-1, m]/n + m*a[n-1, m-1]/n; a[n_, m_] /; n < m := 0; a[n_, 0] = 0; a[1, 1] = 1; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Apr 26 2011, after given formula *)

PROG

(Haskell)

a048966 n k = a048966_tabl !! (n-1) !! (k-1)

a048966_row n = a048966_tabl !! (n-1)

a048966_tabl = [1] : f 2 [1] where

f x xs = ys : f (x + 1) ys where

ys = map (flip div x) $ zipWith (+)

(map (* 3) $ zipWith (*) (map (3 * (x - 1) -) [1..]) (xs ++ [0]))

(zipWith (*) [1..] ([0] ++ xs))

-- Reinhard Zumkeller, Feb 19 2014

CROSSREFS

Cf. A034000, A049213, A049223, A049224. a(n, 1)= A025748(n), a(n, 1)= 3^(n-1)*2*A034000(n-1)/n!, n >= 2. Row sums = A025756.