OFFSET
1,3
COMMENTS
This scaled Stirling2 array will be called s2_{3,2}(n,m).
The sequence of row lengths is [1,3,5,7,...]=A005408(n-1).
The generating function for the sequence from column nr. 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).
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.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000 (rows 1 <= n <= 100, flattened.)
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
W. Lang, First 8 rows.
FORMULA
EXAMPLE
[1]; [1,3,2]; [1,7,16,15,5]; [1,12,51,105,114,63,14]; ...
MATHEMATICA
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. *)
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Dec 23 2003
STATUS
approved