A171532 Coefficients of partition Hermite-Eulerian polynomials: p(x,n)= If[n == 0, 1, HermiteH[n, x]*Sum[Eulerian[n-1, k-1]*x^(k - 1), {k, 1, n}]/2^Floor[n/2]]

1, 0, 2, -1, -1, 2, 2, 0, -6, -24, -2, 16, 4, 3, 33, 21, -129, -128, 32, 44, 4, 0, 30, 780, 1940, -260, -2602, -832, 488, 208, 8, -15, -855, -4440, 600, 26265, 23745, -12982, -17574, -1004, 2356, 456, 8, 0, -210, -25200, -249690, -456960, 249942, 969360

Offset

0,3

Comments

Row sums are:

{1, 2, 2, -12, -120, -240, 16560, 292320, -4152960, -243129600, 932601600,...}

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..49.

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

Formula

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

Example

{1},
{0, 2},
{-1, -1, 2, 2},
{0, -6, -24, -2, 16, 4},
{3, 33, 21, -129, -128, 32, 44, 4},
{0, 30, 780, 1940, -260, -2602, -832, 488, 208, 8},
{-15, -855, -4440, 600, 26265, 23745, -12982, -17574, -1004, 2356, 456, 8},
{0, -210, -25200, -249690, -456960, 249942, 969360, 299938, -353568, -180612, 18496, 18888, 1920, 16},
{105, 25935, 449925, 1432515, -1965285, -12461715, -9488129, 9458617, 11950864, 110576, -3222488, -710888, 194576, 68464, 3952, 16},
{0, 1890, 948780, 27604080, 164232180, 221577804, -276419052, -715414464, -177220116, 390281954, 213482816, -45328432, -46481248, -3413104, 2534336, 466880, 16064, 32},
{-945, -957285, -45199350, -420583590, -786209130, 3050516070, 11349909900, 6607322820, -11968740405, -13764935025, 843893042, 5673703786, 1339479880, -687787736, -280701392, 7491568, 13836784, 1530160, 32416, 32}

Mathematica

Clear[A, p, n, k]
m = 1;
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: A240871 A236765 A171531 * A171533 A115236 A307777

Adjacent sequences: A171529 A171530 A171531 * A171533 A171534 A171535

Keyword

sign,uned

Author

Roger L. Bagula, Dec 11 2009

Status

approved