A112694 -id:A112694 - cp's OEIS frontend

A112692 Coefficient array of numerator polynomials of o.g.f.s (rising powers) for the columns of triangle A008517 (second-order Eulerian numbers).

1, 3, -1, -6, 6, -9, -70, 163, -42, -72, 30, -123, -1110, 8440, -18244, 2423, 43036, -53172, 11232, 8640, 90, -792, -7425, 137760, -771911, 1624514, 2262109, -21114844, 51074797, -54783526, 6214788, 45596664, -40513824, 7309440, 3110400, 630, -10278, -86841, 3685605, -41159454

Offset: 0

Wolfdieter Lang, Oct 14 2005

The sequence of row lengths is A000217 (triangular numbers): [1, 3, 6, 10, 15, 21,..].

The o.g.f. of the k-th column sequence of triangle A008517(n,k), n>=k>=1, is (2^floor(k/2))*(x^k)*p(k,x)/product((1-j*x)^(k+1-j),j=1..k), k>=2, with the row polynomials p(k,x):= sum(a(k-2,m)*x^m,m=0..(k*(k-1)/2)-1).

			Rows: [1]; [3,-1,-6]; [6,-9,-70,163,-42,-72];...
The k=3, offset 3, column sequence [6,58,328,..] of A008517 has o.g.f. 2*(x^3)*(3-x-6*x^2)/product((1-j*x)^(4-j),j=1..3).

W. Lang, First ten rows.

Row sums A112693. Unsigned row sums A112694.

A112693 Row sums of array A112692.

Original entry on oeis.org

1, -4, -48, 2304, 552960, -796262400, -8026324992000, 647242847354880000, 469742968896277708800000, -3409206571061625099386880000000, -272169233711505353534412423168000000000

Offset: 0

Wolfdieter Lang, Oct 14 2005

Unsigned row sums A112694.

a(n)=sum(A112692(k, m), m=0..((k+1)*(k+2)/2)-1), k>=0.

Let A(x)=sum(k>=0, |a(k)|*x^k ), then A(x)= G(0)/(4*x)- 1/(2*x), where G(k)= 1 + 1/(1 - 2*x*(2*k+1)!/(2*x*(2*k+1)! + 1/(1 + 1/(1 - 2*x*(2*k+2)!/(2*x*(2*k+2)! + 1/G(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Jul 10 2013

cp's OEIS Frontend

A112692 Coefficient array of numerator polynomials of o.g.f.s (rising powers) for the columns of triangle A008517 (second-order Eulerian numbers).

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A112693 Row sums of array A112692.

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