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%I #17 May 14 2016 03:54:03
%S 1,4,15,84,470,3552,26796,255840,2435184,28114560,323405280,
%T 4380445440,59105255040,918796677120,14228252640000,249644312064000,
%U 4363865549568000,85297521899520000,1661265370695168000
%N a(n) = (n+1+(-1)^n)*A024167(n), related to alternating harmonic sums.
%C 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 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 The table of coefficients of P() with 2n coefficients per row starts:
%C 2, 1;
%C 3, 3/2, 1, 1/2;
%C 4, 2, 4/3, 1, 1/2, 1/3;.. .
%C Rows multiplied by n!, the table becomes Q():
%C 2, 1;
%C 6, 3, 2, 1;
%C 24, 12, 8, 6, 3, 2;
%C 120, 60, 40, 30, 24, 12, 8, 6;
%C 720, 360, 240, 180, 144,...
%C 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 The row sums of Q are (n+2)*A000254(n)= 3, 12, 55, 300...
%C Adding the alternating and ordinary row sums yields the sequence 4, 16, 70, 384....
%C The sequence of sums of antidiagonals in the Q table starts 2, 6+1=7, 24+3=27, 120+12+1=134.
%H G. C. Greubel, <a href="/A130679/b130679.txt">Table of n, a(n) for n = 1..250</a>
%H Jean-François Alcover, <a href="/A130679/a130679.pdf">Roots of P(2n-1,x) lie close to a circle</a> (example)
%H 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 a(n) = n!*(n+1+(-1)^n)*A058313(n)/A058312(n). - _R. J. Mathar_, Jul 28 2008
%e a(1) = 2-1.
%e a(2) = 6-3+2-1.
%e a(3) = 24-12+8-6+3-2.
%t 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 nonn
%O 1,2
%A _Paul Curtz_, Jun 29 2007
%E Edited and extended by _R. J. Mathar_, Jul 28 2008
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