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

1, 1, 3, 2, 1, 7, 16, 15, 5, 1, 12, 51, 105, 114, 63, 14, 1, 18, 118, 396, 771, 910, 644, 252, 42, 1, 25, 230, 1110, 3235, 6083, 7580, 6240, 3270, 990, 132, 1, 33, 402, 2600, 10365, 27483, 50464, 65331, 59625, 37620, 15642, 3861, 429, 1, 42, 651, 5390, 27825, 97188

Offset: 1

Author

Wolfdieter Lang, Dec 23 2003

Keywords

• nonn
• easy
• tabf

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 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).

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.

Examples

			Triangle begins:
  [1];
  [1,3,2];
  [1,7,16,15,5];
  [1,12,51,105,114,63,14];
  ...
		

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.
• Wolfdieter Lang, First 8 rows.

Crossrefs

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.

The column sequences (without leading zeros) are: A000012 (powers of 1), A055998, A090453-4, A091026-7, etc.

Cf. A090442 (row sums). The alternating row sums are 0 except for row n=1 which gives 1.

Programs

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. *)

Formula

a(n, m) = (m!/((n+1)!*n!))*A078740(n, m), n>=1, 2<= m <=2*n.

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).