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A090452 Scaled array A078740 ((3,2)-Stirling2).

%I A090452 #23 Apr 08 2025 08:45:36
%S A090452 1,1,3,2,1,7,16,15,5,1,12,51,105,114,63,14,1,18,118,396,771,910,644,
%T A090452 252,42,1,25,230,1110,3235,6083,7580,6240,3270,990,132,1,33,402,2600,
%U A090452 10365,27483,50464,65331,59625,37620,15642,3861,429,1,42,651,5390,27825,97188
%N A090452 Scaled array A078740 ((3,2)-Stirling2).
%C A090452 This scaled Stirling2 array will be called s2_{3,2}(n,m).
%C A090452 The sequence of row lengths is [1,3,5,7,...]=A005408(n-1).
%C A090452 The generating function for the sequence from column no. m is G(m,x)=(x^ceiling(m/2))*P(m,x)/(1-x)^(2*m-3) with the row polynomials of array A091029(m,k).
%C A090452 The generating functions of the column sequences obey the hypergeometric differential-difference eq.:x*(1-x)*G''(m,x) + 2*(1-m*x)*G'(m,x) - m*(m-1)*G(m,x) = 2*m*x*G'(m-1,x) + 2*m*(m-1)*G(m-1,x) + m*(m-1)*G(m-2,x), m>=3; with G(2,x)=x/(1-x) and G(1,x)=0. The primes denote differentiation w.r.t. x.
%H A090452 Michael De Vlieger, <a href="/A090452/b090452.txt">Table of n, a(n) for n = 1..10000</a> (rows 1 <= n <= 100, flattened.)
%H A090452 Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
%H A090452 Wolfdieter Lang, <a href="/A090452/a090452.txt">First 8 rows</a>.
%F A090452 a(n, m) = (m!/((n+1)!*n!))*A078740(n, m), n>=1, 2<= m <=2*n.
%F A090452 Recursion: a(n, m) = ((n+m-1)*(n+m-2)*a(n-1, m)+2*(n+m-2)*m*a(n-1, m-1)+m*(m-1)*a(n-1, m-2))/((n+1)*n), n>=2, 2<=m<=2*n, a(1, 2)=1, a(n, 0) := 0, a(n, 1) := 0 (from the recursion of array A078740).
%e A090452 Triangle begins:
%e A090452   [1];
%e A090452   [1,3,2];
%e A090452   [1,7,16,15,5];
%e A090452   [1,12,51,105,114,63,14];
%e A090452   ...
%t A090452 Table[(-1)^m*m!*HypergeometricPFQ[{2 - m, n + 1, n + 2}, {2, 3}, 1]/(2 (m - 2)!), {n, 8}, {m, 2, 2 n}] // Flatten (* _Michael De Vlieger_, Nov 21 2019, after _Jean-François Alcover_ at A078740. *)
%Y A090452 a(n, 2*n)=A000108(n) (Catalan), n>=1, a(n, 2*n-1)=3*A002054(n-1), n>=2, a(n, 2*n-2)=A091031(n), n>=2.
%Y A090452 The column sequences (without leading zeros) are: A000012 (powers of 1), A055998, A090453-4, A091026-7, etc.
%Y A090452 Cf. A090442 (row sums). The alternating row sums are 0 except for row n=1 which gives 1.
%K A090452 nonn,easy,tabf
%O A090452 1,3
%A A090452 _Wolfdieter Lang_, Dec 23 2003