A171533 - OEIS

%I #4 Nov 07 2014 05:32:22

%S 1,0,2,-1,-1,2,2,0,-6,-36,-2,24,4,3,69,57,-273,-272,80,92,4,0,30,2280,

%T 6860,-760,-9162,-2432,1800,608,8,-15,-3555,-25140,-3900,147765,

%U 137145,-79582,-98934,-764,13396,1896,8,0,-210,-151620,-2213610,-4641840

%N Coefficients of partition Hermite-MacMahon polynomials: p(x,n)= If[n == 0, 1, HermiteH[n, x]*Sum[MacMahon[n-1, k-1]*x^(k - 1), {k, 1, n}]/2^Floor[n/2]]

%C Row sums are:

%C {1, 2, 2, -16, -240, -768, 88320, 2672640, -66447360, -6915686400, 47749201920,...}

%C These polynomials are suggested by Gaussian limit of the Eulerian numbers in Analytic Combinatorics.

%C A quantum polynomial like:

%C p(x,n,m)= If[n == 0, 1, HermiteH[m, x]*Sum[Binomial[n-1, k-1]*x^(k - 1), {k, 1, n}]/2^Floor[n/2]]

%C might be a wave function for a system of Hamiltonian equations.

%C I set the Mathematica up so the Eulerian numbers were included in the general form.

%H Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 695.

%F p(x,n)= If[n == 0, 1, HermiteH[n, x]*Sum[MacMahon[n-1, k-1]*x^(k - 1), {k, 1, n}]/2^Floor[n/2]]

%e {1},

%e {0, 2},

%e {-1, -1, 2, 2},

%e {0, -6, -36, -2, 24, 4},

%e {3, 69, 57, -273, -272, 80, 92, 4},

%e {0, 30, 2280, 6860, -760, -9162, -2432, 1800, 608, 8},

%e {-15, -3555, -25140, -3900, 147765, 137145, -79582, -98934, -764, 13396, 1896, 8},

%e {0, -210, -151620, -2213610, -4641840, 2213862, 9617244, 2656642, -3641272, -1602116, 255472, 168520, 11552, 16},

%e {105, 228795, 6368145, 25440555, -23680125, -209967975, -166986869, 166727449, 202749808, -7192048, -55377080, -9430760, 3667472, 970288, 34864, 16},

%e {0, 1890, 12383280, 626741640, 4664172240, 7164455004, -7808843952, -21932529768, -5001731280, 12274911266, 6051172608, -1679406832, -1332183424, -41405040, 75755264, 10611008, 209664, 32},

%e {-945, -18590985, -1659731850, -20328123690, -48998162430, 139296892770, 613311715800, 380874591720, -660633807105, -748616078025, 75243899642, 312084016906, 59238930280, -40430975576, -13309362512, 956665648, 680490544, 56202160, 629536, 32}

%t Clear[A, p, n, k]

%t m = 2;

%t A[n_, 1] := 1 A[n_, n_] := 1

%t A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]

%t a = Table[A[n, k], {n, 10}, {k, n}]

%t p[x_, n_] := If[n == 0, 1, HermiteH[ n, x]*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/2^Floor[n/2]]

%t b = Table[CoefficientList[p[x, n], x], {n, 0, 10}]

%t Flatten[b]

%Y A171229

%K sign,uned

%O 0,3

%A _Roger L. Bagula_, Dec 11 2009