A136264 -id:A136264 - cp's OEIS frontend

A115046 G.f.: (1+x^2)^2(x^4-6x^3+1)/(x^2-1)^4.

1, 0, 6, -6, 20, -36, 50, -114, 104, -264, 190, -510, 316, -876, 490, -1386, 720, -2064, 1014, -2934, 1380, -4020, 1826, -5346, 2360, -6936, 2990, -8814, 3724, -11004, 4570, -13530, 5536, -16416, 6630, -19686, 7860, -23364, 9234, -27474, 10760, -32040, 12446, -37086, 14300, -42636, 16330

Offset: 0

Roger L. Bagula, Feb 28 2006

This sequence arose from an arithmetic error when computing A136264.

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

a(n)= -n/12+3/4-n^3/6+3n^2/4+(-1)^n*(-3n^2/4+29n/12-3/4+n^3/3), n>0. [From R. J. Mathar, Mar 15 2009]

Edited by N. J. A. Sloane, Dec 21 2006 and Apr 19 2008

A137514 A triangular sequence from umbral calculus expansion of Simon Plouffe's rational polynomial for A002890: p(x,t) = exp(xt)(1 - 6t + 9t^2 - 4t^3 + t^4)/(4t - 1)/(2*t - 1).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 12, 6, 0, 1, 120, 48, 12, 0, 1, 1680, 600, 120, 20, 0, 1, 31680, 10080, 1800, 240, 30, 0, 1, 766080, 221760, 35280, 4200, 420, 42, 0, 1, 22579200, 6128640, 887040, 94080, 8400, 672, 56, 0, 1, 778014720, 203212800, 27578880, 2661120, 211680, 15120, 1008, 72, 0, 1

Offset: 1

Roger L. Bagula, Apr 23 2008

Row sums:
{1, 1, 3, 19, 181, 2421, 43831, 1027783, 29698089, 1011695401, 39319102891}
The t's here are actually Sqrt[] of the variables that give Gamma(1,t) in the Hill reference and is the expansion of Plouffe's rational polynomial for A002890. So this result is related closely to Hill's Gamma(x,y) and seems to be a generalization of the A002890 polynomial.

			Triangle begins:
  {1},
  {0, 1},
  {2, 0, 1},
  {12, 6, 0, 1},
  {120, 48, 12, 0, 1},
  {1680, 600, 120, 20, 0, 1},
  {31680, 10080, 1800, 240, 30, 0, 1},
  {766080, 221760, 35280, 4200, 420, 42, 0, 1},
  {22579200, 6128640, 887040, 94080, 8400, 672, 56, 0, 1},
  {778014720, 203212800, 27578880, 2661120, 211680, 15120, 1008, 72, 0, 1},
  ...

Terrel L. Hill, Statistical Mechanics: Principles and Selected Applications, Dover, New York, 1956, page 336 ff

Cf. A002890, A136264.

Clear[p, f, g] p[t_] = Exp[x*t]*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1); Table[ ExpandAll[n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}] a = Table[ CoefficientList[n!*SeriesCoefficient[; FullSimplify[Series[p[t], {t, 0, 30}]], n], x], {n, 0, 10}]; Flatten[a]

p(x,t) = exp(x*t)*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1) = Sum_{n>=0} P(x,n)*t^n/n!; out_n,m=n!*Coefficients(P(x,n)).

A137785 Triangular sequence of coefficients of the expansion of p(x,t) = exp(xt)(1 + t^2)^2/(t*(1 - t^2)).

Original entry on oeis.org

0, 1, 6, 0, 1, 0, 18, 0, 1, 96, 0, 36, 0, 1, 0, 480, 0, 60, 0, 1, 2880, 0, 1440, 0, 90, 0, 1, 0, 20160, 0, 3360, 0, 126, 0, 1, 161280, 0, 80640, 0, 6720, 0, 168, 0, 1, 0, 1451520, 0, 241920, 0, 12096, 0, 216, 0, 1, 14515200, 0, 7257600, 0, 604800, 0, 20160, 0, 270, 0, 1

Offset: 1

Roger L. Bagula, Apr 28 2008

			{0, 1},
{6, 0, 1},
{0, 18, 0, 1},
{96, 0, 36, 0, 1},
{0, 480, 0, 60, 0, 1},
{2880, 0, 1440, 0, 90, 0, 1},
{0, 20160, 0, 3360, 0, 126, 0, 1},
{161280, 0, 80640, 0, 6720, 0, 168, 0, 1},
{0, 1451520, 0, 241920, 0, 12096, 0, 216, 0, 1},
{14515200, 0, 7257600, 0, 604800, 0, 20160, 0, 270, 0, 1},
{0, 159667200, 0, 26611200, 0, 1330560, 0, 31680, 0, 330, 0, 1}

The Beauty of Fractals, Springer-Verlag, New York, 1986, editors Peitgen and Richter, pages 153
Terrell Hill, Statistical Mechanics, Dover, 1987, page 329 ff

Cf. A136264.

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

cp's OEIS Frontend

A115046 G.f.: (1+x^2)^2(x^4-6x^3+1)/(x^2-1)^4.

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A137514 A triangular sequence from umbral calculus expansion of Simon Plouffe's rational polynomial for A002890: p(x,t) = exp(xt)(1 - 6t + 9t^2 - 4t^3 + t^4)/(4t - 1)/(2*t - 1).

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A137785 Triangular sequence of coefficients of the expansion of p(x,t) = exp(xt)(1 + t^2)^2/(t*(1 - t^2)).

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cp's OEIS Frontend

A115046 G.f.: (1+x^2)^2*(x^4-6*x^3+1)/(x^2-1)^4.

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Author

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Extensions

A137514 A triangular sequence from umbral calculus expansion of Simon Plouffe's rational polynomial for A002890: p(x,t) = exp(x*t)*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1).

Views

Author

Keywords

Comments

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References

Crossrefs

Programs

Mathematica

Formula

A137785 Triangular sequence of coefficients of the expansion of p(x,t) = exp(x*t)*(1 + t^2)^2/(t*(1 - t^2)).

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

A115046 G.f.: (1+x^2)^2(x^4-6x^3+1)/(x^2-1)^4.

A137514 A triangular sequence from umbral calculus expansion of Simon Plouffe's rational polynomial for A002890: p(x,t) = exp(xt)(1 - 6t + 9t^2 - 4t^3 + t^4)/(4t - 1)/(2*t - 1).

A137785 Triangular sequence of coefficients of the expansion of p(x,t) = exp(xt)(1 + t^2)^2/(t*(1 - t^2)).