A123243 - OEIS

%I #15 Jun 13 2021 17:14:11

%S 1,1,1,2,1,2,3,3,3,1,12,16,17,18,6,12,28,45,63,69,69,57,41,24,6,120,

%T 268,434,613,672,684,570,410,240,60,120,388,822,1435,2107,2791,3361,

%U 3771,3891,3683,3249,2636,1964,1280,710,300,60,1680,5312,11240,19656,28885

%N Triangular array: odd: p(k, x) = 2*x*p(k-1, x) + (1-x2)*p(k-2, x), even: p(k, x) = (Sum_{m=0..k} x^m)*p(k-1, x).

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

%D S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, 1960, page 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 odd: p(k, x) = 2*x*p(k-1, x) + (1-x2)*p(k-2, x);

%F even: p(k, x) = (Sum_{m=0..k} x^m)*p(k-1, x)

%e {1},

%e {1, 1},

%e {2, 1},

%e {2, 3, 3, 3, 1},

%e {12, 16, 17, 18, 6},

%e {12, 28, 45, 63, 69, 69, 57, 41, 24, 6},

%e {120, 268, 434, 613, 672, 684, 570, 410, 240, 60},

%e {120, 388, 822, 1435, 2107, 2791, 3361, 3771, 3891, 3683, 3249, 2636, 1964, 1280, 710, 300, 60}

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

%t p[k_, x_] := p[k, x] = If[Mod[k, 2] == 0, 2*(k - 1)*p[k - 1, x] - x*p[k - 2,x], Sum[x^m, {m, 0, k}]*p[k - 1, x]];

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

%Y Cf. A008302.

%K nonn,uned,tabf

%O 1,4

%A _Roger L. Bagula_, Oct 07 2006