A048966 - OEIS

%I #30 Dec 24 2016 09:31:00

%S 1,3,1,15,6,1,90,39,9,1,594,270,72,12,1,4158,1953,567,114,15,1,30294,

%T 14580,4482,1008,165,18,1,227205,111456,35721,8667,1620,225,21,1,

%U 1741905,867834,287199,73656,15075,2430,294,24,1,13586859,6857136,2328183,623106,136323,24354,3465,372,27,1

%N A convolution triangle of numbers obtained from A025748.

%C A generalization of the Catalan triangle A033184.

%H Reinhard Zumkeller, <a href="/A048966/b048966.txt">Rows n = 1..125 of triangle, flattened</a>

%H W. Lang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

%F 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.

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

%F 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

%e Triangle begins:

%e      1;

%e      3,    1;

%e     15,    6,    1;

%e     90,   39,    9,    1;

%e    594,  270,   72,   12,    1;

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

%t 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 *)

%o (Haskell)

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

%o a048966_row n = a048966_tabl !! (n-1)

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

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

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

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

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

%o -- _Reinhard Zumkeller_, Feb 19 2014

%Y 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.

%K easy,nonn,tabl,nice

%O 1,2

%A _Wolfdieter Lang_