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 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]] 0
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 19 02:20 EDT 2024. Contains 376003 sequences. (Running on oeis4.)