A123242 - OEIS

%I #19 Jun 12 2024 03:33:30

%S 1,1,1,1,2,1,4,7,3,1,10,14,4,-1,8,76,105,29,-8,1,26,165,204,43,-20,1,

%T 12,304,1904,2343,487,-232,12,1,50,772,3986,4564,750,-506,44,-1,16,

%U 788,12048,61872,70681,11513,-7864,692,-16,1,82,2347,28032,127536,138126,17956,-16434,1889,-76,1

%N Irregular triangle: p(k, x) = 2*x*p(k-1, x) + (1 - x^2)*p(k-2, x) for even k, p(k, x) = 2*(k-1)*p(k-1, x) - x*p(k-2, x) for odd k.

%D E. S. R. Gopal, Specific Heats at Low Temperatures, Plenum Press, New York, 1966, pp. 36-40.

%D S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, 1960, p. 110.

%H B. H. Margolius, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/MARGOLIUS/inversions.html">Permutations with inversions</a>, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.

%F p(k, x) = 2*(k - 1)*p(k - 1, x) - x*p(k - 2, x) if k is odd,

%F           2*x*p(k - 1, x) + (1 - x^2)*p(k - 2, x) if k is even.

%e Triangle starts:

%e   1;

%e   1,  1;

%e   1,  2,   1;

%e   4,  7,   3;

%e   1, 10,  14,   4, -1;

%e   8, 76, 105,  29, -8;

%e   1, 26, 165, 204, 43, -20, 1;

%t p[0, x] = 1; p[1, x] = x + 1;

%t p[k_, x_] := p[k, x] = If[Mod[k, 2] == 1, 2*(k - 1)*p[k - 1, x] - x*p[k - 2,x], 2*x*p[k - 1, x] + (1 - x^2)*p[k - 2, x]];

%t w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]

%K uned,tabf,sign

%O 1,5

%A _Roger L. Bagula_, Oct 07 2006