cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A171246 Triangle read by rows: T(n,k) = 1 + floor(n!/2^((k - n/2)^2 + 1)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 7, 13, 7, 1, 1, 13, 51, 51, 13, 1, 1, 23, 181, 361, 181, 23, 1, 1, 34, 530, 2120, 2120, 530, 34, 1, 1, 40, 1261, 10081, 20161, 10081, 1261, 40, 1, 1, 38, 2384, 38144, 152573, 152573, 38144, 2384, 38, 1
Offset: 0

Views

Author

Roger L. Bagula, Dec 06 2009

Keywords

Examples

			Triangle begins as:
   1;
   1,  1;
   1,  2,   1;
   1,  3,   3,    1;
   1,  7,  13,    7,    1;
   1, 13,  51,   51,   13,   1;
   1, 23, 181,  361,  181,  23,  1;
   1, 34, 530, 2120, 2120, 530, 34, 1;

Links

Crossrefs

Cf. A171229.

Programs

  • Magma
    [[1 +Floor(Factorial(n)/2^((k - n/2)^2 +1)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 11 2019
  • Mathematica
    T[n_, k_]:= 1 +Floor[n!*2^(-(k-n/2)^2 -1)]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
  • PARI
    {T(n,k) = 1 + floor(n!/2^((k - n/2)^2 +1))}; \\ G. C. Greubel, Apr 11 2019
  • Sage
    [[1 + floor(factorial(n)/2^((k-n/2)^2 +1)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 11 2019

Formula

T(n,k) = 1 + floor(n!/2^((k - n/2)^2 +1)).

Extensions

Edited by G. C. Greubel, Apr 11 2019

A171531 Irregular triangle read by rows: first row is 1, n-th row (n > 0) consists of the coefficients in the expansion of H(x;n)*(x + 1)^(n - 1)/2^floor(n/2), where H(x;n) is the Hermite polynomial of order n.

Original entry on oeis.org

1, 0, 2, -1, -1, 2, 2, 0, -6, -12, -2, 8, 4, 3, 9, -3, -33, -32, 0, 12, 4, 0, 30, 120, 140, -40, -202, -128, 8, 32, 8, -15, -75, -60, 300, 765, 585, -142, -470, -220, 20, 40, 8, 0, -210, -1260, -2730, -1680, 2982, 6132, 3586, -744, -1860, -688, 72, 96, 16, 105, 735, 1365
Offset: 0

Views

Author

Roger L. Bagula, Dec 11 2009

Keywords

Examples

			Triangle begins:
    1;
    0,   2;
   -1,  -1,   2,   2;
    0,  -6, -12,  -2,   8,    4;
    3,   9,  -3, -33, -32,    0,   12,    4;
    0,  30, 120, 140, -40, -202, -128,    8,   32,  8;
  -15, -75, -60, 300, 765,  585, -142, -470, -220, 20, 40, 8;
   ... reformatted. - _Franck Maminirina Ramaharo_, Oct 02 2018

Links

Crossrefs

Cf. A007318, A060821, A171229, A171631.

Programs

  • Mathematica
    Join[{1},Table[CoefficientList[HermiteH[n,x]*(x + 1)^(n - 1)/2^Floor[n/2], x], {n, 1, 12}]]//Flatten

Extensions

Edited and new name by Franck Maminirina Ramaharo, Oct 02 2018

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]].

Original entry on oeis.org

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

Views

Author

Roger L. Bagula, Dec 11 2009

Keywords

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.

Examples

			{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}

Links

Crossrefs

A171229

Programs

  • 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]

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]]

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]].

Original entry on oeis.org

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

Views

Author

Roger L. Bagula, Dec 11 2009

Keywords

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.

Examples

			{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}

Links

Crossrefs

A171229

Programs

  • 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]

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]]
Showing 1-4 of 4 results.

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