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 1-3 of 3 results.

A293561 Column 3 of A142249.

Original entry on oeis.org

3, 19, 80, 286, 945, 2997, 9294, 28456, 86471, 261559, 788892, 2375010, 7141581, 21457705, 64438874, 193447948, 580606275, 1742343435, 5228079240, 15686335270, 47063200553, 141197990749, 423610749990, 1270865804976, 3812664524415, 11438127791647
Offset: 3

Views

Author

Vaclav Kotesovec, Oct 12 2017

Keywords

Crossrefs

Cf. A142249.

Programs

  • Mathematica
    Table[(CoefficientList[Simplify[(1-x)^n * PolyLog[-n, 2, x] / (x*Log[1-x])], x]/.x->1-E)[[3]], {n, 4, 20}]

Formula

a(n) ~ 3^n / 2.

A293562 Column 4 of A142249.

Original entry on oeis.org

4, 65, 566, 3710, 20756, 105299, 500862, 2278936, 10046620, 43280341, 183278902, 766153042, 3171277460, 13027450455, 53203165950, 216290270892, 876183166140, 3539559610905, 14267986165270, 57416955957350, 230750092893524, 926388737476955, 3716144413352126
Offset: 4

Views

Author

Vaclav Kotesovec, Oct 12 2017

Keywords

Crossrefs

Cf. A142249.

Programs

  • Mathematica
    Table[(CoefficientList[Simplify[(1-x)^n * PolyLog[-n, 2, x] / (x*Log[1-x])], x]/.x->1-E)[[4]], {n, 5, 20}]

Formula

a(n) ~ 5 * 4^n / 6.

A293298 Triangle read by rows, a generalization of the Eulerian numbers based on Nielsen's generalized polylogarithm (case m = 3).

Original entry on oeis.org

1, 0, 1, 0, 1, -2, 0, 1, -5, 2, 0, 1, -10, 5, 0, 1, -19, 1, 11, 0, 1, -36, -46, 84, 19, 0, 1, -69, -272, 358, 393, 29, 0, 1, -134, -1149, 916, 4171, 1322, 41, 0, 1, -263, -4237, -191, 31939, 26255, 3841, 55, 0, 1, -520, -14536, -20192, 200252, 348848, 130924, 10280, 71
Offset: 0

Views

Author

Peter Luschny, Oct 11 2017

Keywords

Comments

Examples

			Triangle starts:
{1}
{0, 1}
{0, 1,   -2}
{0, 1,   -5,     2}
{0, 1,  -10,     5}
{0, 1,  -19,     1,   11}
{0, 1,  -36,   -46,   84,    19}
{0, 1,  -69,  -272,  358,   393,    29}
{0, 1, -134, -1149,  916,  4171,  1322,   41}
{0, 1, -263, -4237, -191, 31939, 26255, 3841, 55}
		

Crossrefs

A123125 (m=1), A142249 (m=2 with missing first column), this seq. (m=3).

Programs

  • Mathematica
    npl[n_, m_] := (m-1)! (1 - x)^n PolyLog[-n, m, x];
    A293298Row[0] := {1};
    A293298Row[n_] := CoefficientList[FunctionExpand[npl[n, 3]], x] /. Log[1-x] -> 1;
    Table[A293298Row[n], {n, 0, 10}] // Flatten

Formula

Let p(n, m) = (m - 1)!*(1 - x)^n*PolyLog(-n, m, x) and P(n) the polynomial given by the expansion of p(n, m=3) after replacing log(1 - x) by 1. T(n, k) is the k-th coefficient of P(n).
Showing 1-3 of 3 results.