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

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