1,3

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.

W. Lang, First 8 rows.

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).

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

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.

Sequence in context: A082038 A143774 A158474 this_sequence A110439 A065602 A016648

Adjacent sequences: A090449 A090450 A090451 this_sequence A090453 A090454 A090455

nonn,easy,tabf

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 23 2003