A171533 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]]

1, 0, 2, -1, -1, 2, 2, 0, -6, -36, -2, 24, 4, 3, 69, 57, -273, -272, 80, 92, 4, 0, 30, 2280, 6860, -760, -9162, -2432, 1800, 608, 8, -15, -3555, -25140, -3900, 147765, 137145, -79582, -98934, -764, 13396, 1896, 8, 0, -210, -151620, -2213610, -4641840

Offset

0,3

Comments

Row sums are:

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

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

A quantum polynomial like:

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]]

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

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

Links

Table of n, a(n) for n=0..47.

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 695.

Formula

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]]

Example

{1},
{0, 2},
{-1, -1, 2, 2},
{0, -6, -36, -2, 24, 4},
{3, 69, 57, -273, -272, 80, 92, 4},
{0, 30, 2280, 6860, -760, -9162, -2432, 1800, 608, 8},
{-15, -3555, -25140, -3900, 147765, 137145, -79582, -98934, -764, 13396, 1896, 8},
{0, -210, -151620, -2213610, -4641840, 2213862, 9617244, 2656642, -3641272, -1602116, 255472, 168520, 11552, 16},
{105, 228795, 6368145, 25440555, -23680125, -209967975, -166986869, 166727449, 202749808, -7192048, -55377080, -9430760, 3667472, 970288, 34864, 16},
{0, 1890, 12383280, 626741640, 4664172240, 7164455004, -7808843952, -21932529768, -5001731280, 12274911266, 6051172608, -1679406832, -1332183424, -41405040, 75755264, 10611008, 209664, 32},
{-945, -18590985, -1659731850, -20328123690, -48998162430, 139296892770, 613311715800, 380874591720, -660633807105, -748616078025, 75243899642, 312084016906, 59238930280, -40430975576, -13309362512, 956665648, 680490544, 56202160, 629536, 32}

Mathematica

Clear[A, p, n, k]
m = 2;
A[n_, 1] := 1 A[n_, n_] := 1
A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]
a = Table[A[n, k], {n, 10}, {k, n}]
p[x_, n_] := If[n == 0, 1, HermiteH[ n, x]*Sum[a[[n, k]]*x^(k - 1), {k, 1, n}]/2^Floor[n/2]]
b = Table[CoefficientList[p[x, n], x], {n, 0, 10}]
Flatten[b]

Crossrefs

A171229

Sequence in context: A236765 A171531 A171532 * A115236 A307777 A365661

Adjacent sequences: A171530 A171531 A171532 * A171534 A171535 A171536

Keyword

sign,uned

Author

Roger L. Bagula, Dec 11 2009

Status

approved