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A130679 a(n) = (n+1+(-1)^n)*A024167(n), related to alternating harmonic sums. 4
1, 4, 15, 84, 470, 3552, 26796, 255840, 2435184, 28114560, 323405280, 4380445440, 59105255040, 918796677120, 14228252640000, 249644312064000, 4363865549568000, 85297521899520000, 1661265370695168000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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.

FORMULA

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

EXAMPLE

a(1) = 2-1.

a(2) = 6-3+2-1.

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

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 *)

CROSSREFS

Sequence in context: A117927 A143340 A151379 * A243048 A107874 A237627

Adjacent sequences:  A130676 A130677 A130678 * A130680 A130681 A130682

KEYWORD

nonn

AUTHOR

Paul Curtz, Jun 29 2007

EXTENSIONS

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

STATUS

approved

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Last modified March 22 22:17 EDT 2017. Contains 283901 sequences.