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

%I A130679 #17 May 14 2016 03:54:03
%S A130679 1,4,15,84,470,3552,26796,255840,2435184,28114560,323405280,
%T A130679 4380445440,59105255040,918796677120,14228252640000,249644312064000,
%U A130679 4363865549568000,85297521899520000,1661265370695168000
%N A130679 a(n) = (n+1+(-1)^n)*A024167(n), related to alternating harmonic sums.
%C A130679 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).
%C A130679 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.
%C A130679 The table of coefficients of P() with 2n coefficients per row starts:
%C A130679 2, 1;
%C A130679 3, 3/2, 1, 1/2;
%C A130679 4, 2, 4/3, 1, 1/2, 1/3;.. .
%C A130679 Rows multiplied by n!, the table becomes Q():
%C A130679 2, 1;
%C A130679 6, 3, 2, 1;
%C A130679 24, 12, 8, 6, 3, 2;
%C A130679 120, 60, 40, 30, 24, 12, 8, 6;
%C A130679 720, 360, 240, 180, 144,...
%C A130679 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.
%C A130679 The row sums of Q are (n+2)*A000254(n)= 3, 12, 55, 300...
%C A130679 Adding the alternating and ordinary row sums yields the sequence 4, 16, 70, 384....
%C A130679 The sequence of sums of antidiagonals in the Q table starts 2, 6+1=7, 24+3=27, 120+12+1=134.
%H A130679 G. C. Greubel, <a href="/A130679/b130679.txt">Table of n, a(n) for n = 1..250</a>
%H A130679 Jean-François Alcover, <a href="/A130679/a130679.pdf">Roots of P(2n-1,x) lie close to a circle</a> (example)
%H A130679 P. Curtz, <a href="http://www.numdam.org/item?id=ASENS_1980_4_13_3_293_0">Stabilité locale des systèmes quadratiques</a>, Ann. Sc. Ec. Norm. Sup., vol. 13 no. 3 (1980) 293-302.
%F A130679 a(n) = n!*(n+1+(-1)^n)*A058313(n)/A058312(n). - _R. J. Mathar_, Jul 28 2008
%e A130679 a(1) = 2-1.
%e A130679 a(2) = 6-3+2-1.
%e A130679 a(3) = 24-12+8-6+3-2.
%t A130679 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 *)
%K A130679 nonn
%O A130679 1,2
%A A130679 _Paul Curtz_, Jun 29 2007
%E A130679 Edited and extended by _R. J. Mathar_, Jul 28 2008