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A048966 A convolution triangle of numbers obtained from A025748.

%I A048966 #30 Dec 24 2016 09:31:00
%S A048966 1,3,1,15,6,1,90,39,9,1,594,270,72,12,1,4158,1953,567,114,15,1,30294,
%T A048966 14580,4482,1008,165,18,1,227205,111456,35721,8667,1620,225,21,1,
%U A048966 1741905,867834,287199,73656,15075,2430,294,24,1,13586859,6857136,2328183,623106,136323,24354,3465,372,27,1
%N A048966 A convolution triangle of numbers obtained from A025748.
%C A048966 A generalization of the Catalan triangle A033184.
%H A048966 Reinhard Zumkeller, <a href="/A048966/b048966.txt">Rows n = 1..125 of triangle, flattened</a>
%H A048966 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 A048966 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 A048966 G.f. for m-th column: ((1-(1-9*x)^(1/3))/3)^m.
%F A048966 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 A048966 Triangle begins:
%e A048966      1;
%e A048966      3,    1;
%e A048966     15,    6,    1;
%e A048966     90,   39,    9,    1;
%e A048966    594,  270,   72,   12,    1;
%e A048966   4158, 1953,  567,  114,   15,    1;
%t A048966 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 A048966 (Haskell)
%o A048966 a048966 n k = a048966_tabl !! (n-1) !! (k-1)
%o A048966 a048966_row n = a048966_tabl !! (n-1)
%o A048966 a048966_tabl = [1] : f 2 [1] where
%o A048966    f x xs = ys : f (x + 1) ys where
%o A048966      ys = map (flip div x) $ zipWith (+)
%o A048966           (map (* 3) $ zipWith (*) (map (3 * (x - 1) -) [1..]) (xs ++ [0]))
%o A048966           (zipWith (*) [1..] ([0] ++ xs))
%o A048966 -- _Reinhard Zumkeller_, Feb 19 2014
%Y A048966 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 A048966 easy,nonn,tabl,nice
%O A048966 1,2
%A A048966 _Wolfdieter Lang_