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A035529 A convolution triangle of numbers obtained from A034171.

%I A035529 #21 Aug 07 2024 09:18:54
%S A035529 1,6,1,42,12,1,315,120,18,1,2457,1134,234,24,1,19656,10458,2673,384,
%T A035529 30,1,160056,95256,28539,5148,570,36,1,1320462,861597,292572,62532,
%U A035529 8775,792,42,1,11003850,7760610,2920347,713664,119565,13770,1050,48,1
%N A035529 A convolution triangle of numbers obtained from A034171.
%C A035529 a(n,1)= A034171(n-1); a(n,m)=: s2(4; n,m), generalizing s2(2; n,m) := A007318(n-1,m-1) (Pascal), s2(3; n,m) := A035324(n,m).
%H A035529 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A035529 a(n+1, m) = 3*(3*n+m)*a(n, m)/(n+1) + m*a(n, m-1)/(n+1), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0; a(1, 1)=1.
%F A035529 G.f. for column m: ((-1+(1-9*x)^(-1/3))/3)^m.
%e A035529 Triangle begins:
%e A035529       1,
%e A035529       6,     1;
%e A035529      42,    12,    1;
%e A035529     315,   120,   18,   1;
%e A035529    2457,  1134,  234,  24,  1;
%e A035529   19656, 10458, 2673, 384, 30, 1;
%e A035529   ...
%t A035529 a[n_, m_] /; n - 1 >= m >= 1 := (m*a[n - 1, m - 1])/n + (3*(m + 3*(n - 1))*a[n - 1, m])/n; a[n_, m_] /; n < m = 0; a[n_, 0] = 0; a[n_, n_] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* _Jean-François Alcover_, Jul 10 2012, from formula *)
%Y A035529 Cf. A034171, A007318, A035324.
%Y A035529 Row sums: A049028(n), n >= 1.
%K A035529 easy,nice,nonn,tabl
%O A035529 1,2
%A A035529 _Wolfdieter Lang_