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 1-2 of 2 results.

A171683 Triangle T(n,k) which contains 4*n!*2^floor((n+1)/2) times the coefficient [t^n x^k] exp(t*x)/(3 + exp(2*t)) in row n, column k.

Original entry on oeis.org

1, -1, 2, -1, -2, 2, 1, -6, -6, 4, 10, 4, -12, -8, 4, 26, 100, 20, -40, -20, 8, -154, 156, 300, 40, -60, -24, 8, -1646, -2156, 1092, 1400, 140, -168, -56, 16, 1000, -13168, -8624, 2912, 2800, 224, -224, -64, 16, 92744, 18000, -118512, -51744, 13104, 10080, 672, -576, -144, 32
Offset: 0

Views

Author

Roger L. Bagula, Dec 15 2009

Keywords

Comments

The bivariate Taylor expansion of exp(t*x)/(3+exp(2*t)) is 1/4 + (x/4-1/8)*t +(-1/16+x^2/8-x/8)*t^2+...
Row n contains the coefficients of [x^k] of the polynomial in front of t^n, multiplied by 4*floor((n+1)/2)*n!.
Row sums are: 1, 1, -1, -7, -2, 94, 266, -1378, -15128, -36344, 839144,...

Examples

			The triangle starts in row n=0, columns 0<=k <=n as
      1;
     -1,      2;
     -1,     -2,       2;
      1,     -6,      -6,      4;
     10,      4,     -12,     -8,     4;
     26,    100,      20,    -40,   -20,     8;
   -154,    156,     300,     40,   -60,   -24,    8;
  -1646,  -2156,    1092,   1400,   140,  -168,  -56,   16;
   1000, -13168,   -8624,   2912,  2800,   224, -224,  -64,   16;
  92744,  18000, -118512, -51744, 13104, 10080,  672, -576, -144, 32;
  ...

Crossrefs

Cf. A000364, A060083, A171684.

Programs

  • Mathematica
    Clear[p, g, m, a];
m = 1;
p[t_] = 2^(m + 1)*Exp[t*x]/(-1 + 2^(m + 1) + Exp[2^m*t]) Table[ FullSimplify[ExpandAll[2^ Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], {n, 0, 10}]
a = Table[CoefficientList[FullSimplify[ExpandAll[2^Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}]
Flatten[a]

Extensions

Number of variables in use reduced from 4 to 2, keyword:tabl added - The Assoc. Eds. of the OEIS, Oct 20 2010

A171685 Triangle T(n,k) which contains 16*n!*2^floor((n+1)/2) times the coefficient [t^n x^k] exp(t*x)/(15 + exp(8*t)) in row n, column k.

Original entry on oeis.org

1, -1, 2, -7, -2, 2, -83, -42, -6, 4, -266, -332, -84, -8, 4, 5666, -2660, -1660, -280, -20, 8, 146762, 33996, -7980, -3320, -420, -24, 8, 3415978, 2054668, 237972, -37240, -11620, -1176, -56, 16, 7599256, 27327824, 8218672, 634592, -74480, -18592, -1568, -64, 16
Offset: 0

Views

Author

Roger L. Bagula, Dec 15 2009

Keywords

Comments

The bivariate Taylor expansion of exp(t*x)/(15+exp(8*t)) is 1/16 + (x/16-1/32)*t +(-7/64+x^2/32 -x/32)*t^2+ (-83/384+x^3/96-7*x/64-x^2/64)*t^3+...
Row n contains the coefficients of the polynomial in front of t^n, multiplied by 16*floor[(n+1)/2]*n!.
Row sums are: 1, 1, -7, -127, -686, 1054, 169022, 5658542, 43685656, -1052651384, -55785840712,....

Examples

			The triangle starts in row n = 0 with columns 0 <= k <= n as
  1;
  -1, 2;
  -7, -2, 2;
  -83, -42, -6, 4;
  -266, -332, -84, -8, 4;
  5666, -2660, -1660, -280, -20, 8;
  146762, 33996, -7980, -3320, -420, -24, 8;
  3415978, 2054668, 237972, -37240, -11620, -1176, -56, 16;
  7599256, 27327824, 8218672, 634592, -74480, -18592, -1568, -64, 16;
  ...

Crossrefs

Cf. A000364, A171684.

Programs

  • Mathematica
    Clear[p, g, m, a];
m = 3;
p[t_] = 2^(m + 1)*Exp[t*x]/(-1 + 2^(m + 1) + Exp[2^m*t])
Table[ FullSimplify[ExpandAll[2^ Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], {n, 0, 10}]
a = Table[CoefficientList[FullSimplify[ExpandAll[2^Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}]
Flatten[a]

Extensions

Number of variables in use reduced from 4 to 2, keyword:tabl added - The Assoc. Eds. of the OEIS, Oct 20 2010
Showing 1-2 of 2 results.

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