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A048966 A convolution triangle of numbers obtained from A025748. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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:

     1;

     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.

Sequence in context: A144815 A065250 A092589 * A297704 A104990 A089463

Adjacent sequences:  A048963 A048964 A048965 * A048967 A048968 A048969

KEYWORD

easy,nonn,tabl,nice

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified April 23 05:42 EDT 2021. Contains 343199 sequences. (Running on oeis4.)