A130679 a(n) = (n+1+(-1)^n)*A024167(n), related to alternating harmonic sums.

1, 4, 15, 84, 470, 3552, 26796, 255840, 2435184, 28114560, 323405280, 4380445440, 59105255040, 918796677120, 14228252640000, 249644312064000, 4363865549568000, 85297521899520000, 1661265370695168000

Offset: 1

Author

Paul Curtz, Jun 29 2007

Keywords

• nonn

Comments

Inspired by a formula in the reference, the study of the singular points of planar differential systems leads to 3 two-dimensional polynomial families, one ordinary (degenerate case, considered in one dimension, see A129326) and two odd (the second, considered in one dimension, see A129587).

The first is in one dimension P(2n-1,x)=(n+1+x^n)*sum_{i=0..n-1} x^i/(i+1), n>=1.

The table of coefficients of P() with 2n coefficients per row starts:

2, 1;

3, 3/2, 1, 1/2;

4, 2, 4/3, 1, 1/2, 1/3;.. .

Rows multiplied by n!, the table becomes Q():

2, 1;

6, 3, 2, 1;

24, 12, 8, 6, 3, 2;

120, 60, 40, 30, 24, 12, 8, 6;

720, 360, 240, 180, 144,...

The sequence gives the alternating row sums of this table Q, positive sign for coefficients in front of even and negative sign for coefficients in front of odd powers of x.

The row sums of Q are (n+2)*A000254(n)= 3, 12, 55, 300...

Adding the alternating and ordinary row sums yields the sequence 4, 16, 70, 384....

The sequence of sums of antidiagonals in the Q table starts 2, 6+1=7, 24+3=27, 120+12+1=134.

Examples

			a(1) = 2-1.
a(2) = 6-3+2-1.
a(3) = 24-12+8-6+3-2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..250
• Jean-François Alcover, Roots of P(2n-1,x) lie close to a circle (example)
• P. Curtz, Stabilité locale des systèmes quadratiques, Ann. Sc. Ec. Norm. Sup., vol. 13 no. 3 (1980) 293-302.

Programs

Mathematica

a[n_] := (1/2)*(n + (-1)^n + 1)*n!*((-1)^n*(HarmonicNumber[(n-1)/2] - HarmonicNumber[n/2]) + Log[4]); Table[a[n] // FullSimplify, {n, 1, 19}] (* Jean-François Alcover, Oct 03 2012 *)

Formula

a(n) = n!*(n+1+(-1)^n)*A058313(n)/A058312(n). - R. J. Mathar, Jul 28 2008

Extensions

Edited and extended by R. J. Mathar, Jul 28 2008