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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]].

%I A171532 #6 Jul 22 2025 07:31:34
%S A171532 1,0,2,-1,-1,2,2,0,-6,-24,-2,16,4,3,33,21,-129,-128,32,44,4,0,30,780,
%T A171532 1940,-260,-2602,-832,488,208,8,-15,-855,-4440,600,26265,23745,-12982,
%U A171532 -17574,-1004,2356,456,8,0,-210,-25200,-249690,-456960,249942,969360
%N 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]].
%C A171532 Row sums are:
%C A171532 {1, 2, 2, -12, -120, -240, 16560, 292320, -4152960, -243129600, 932601600,...}
%C A171532 These polynomials are suggested by Gaussian limit of the Eulerian numbers in Analytic Combinatorics.
%C A171532 A quantum polynomial like:
%C A171532 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 A171532 might be a wave function for a system of Hamiltonian equations.
%C A171532 I set the Mathematica up so the Eulerian numbers were included in the general form.
%H A171532 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, pgae 695.
%F A171532 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]]
%e A171532 {1},
%e A171532 {0, 2},
%e A171532 {-1, -1, 2, 2},
%e A171532 {0, -6, -24, -2, 16, 4},
%e A171532 {3, 33, 21, -129, -128, 32, 44, 4},
%e A171532 {0, 30, 780, 1940, -260, -2602, -832, 488, 208, 8},
%e A171532 {-15, -855, -4440, 600, 26265, 23745, -12982, -17574, -1004, 2356, 456, 8},
%e A171532 {0, -210, -25200, -249690, -456960, 249942, 969360, 299938, -353568, -180612, 18496, 18888, 1920, 16},
%e A171532 {105, 25935, 449925, 1432515, -1965285, -12461715, -9488129, 9458617, 11950864, 110576, -3222488, -710888, 194576, 68464, 3952, 16},
%e A171532 {0, 1890, 948780, 27604080, 164232180, 221577804, -276419052, -715414464, -177220116, 390281954, 213482816, -45328432, -46481248, -3413104, 2534336, 466880, 16064, 32},
%e A171532 {-945, -957285, -45199350, -420583590, -786209130, 3050516070, 11349909900, 6607322820, -11968740405, -13764935025, 843893042, 5673703786, 1339479880, -687787736, -280701392, 7491568, 13836784, 1530160, 32416, 32}
%t A171532 Clear[A, p, n, k]
%t A171532 m = 1;
%t A171532 A[n_, 1] := 1 A[n_, n_] := 1
%t A171532 A[n_, k_] := (m*n - m*k + 1)A[n - 1, k - 1] + (m*k - (m - 1))A[n - 1, k]
%t A171532 a = Table[A[n, k], {n, 10}, {k, n}]
%t A171532 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 A171532 b = Table[CoefficientList[p[x, n], x], {n, 0, 10}]
%t A171532 Flatten[b]
%Y A171532 A171229
%K A171532 sign,uned
%O A171532 0,3
%A A171532 _Roger L. Bagula_, Dec 11 2009