A138133 Triangle read by rows, based on the two-variable g.f. exp(x*t)*(x*(1 - 2*exp(x)) - 2*exp(x))/(1 - exp(t)) (the first of two parts).

0, 2, -4, 0, -1, 6, -6, 0, 0, -8, 24, -16, 0, 2, 0, -60, 120, -60, 0, 0, 48, 0, -480, 720, -288, 0, -40, 0, 840, 0, -4200, 5040, -1680, 0, 0, -1920, 0, 13440, 0, -40320, 40320, -11520, 0, 3024, 0, -60480, 0, 211680, 0, -423360, 362880, -90720, 0, 0, 241920, 0, -1612800, 0, 3386880, 0

Offset

0,2

Comments

A factor of 2*n!*(n+2)! was used to make the coefficients integers.

This is the a(i) part of the Sum[(a(i)+b(i)*Exp(x))*x^i,{i,0,n}] expansion (see A176295 for the exponential part).

Row sums are {-2, -1, 0, 2, 0, -40, 0, 3024, 0, -604800, 0,....}.

References

Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 1965,pp 296-298

Links

Table of n, a(n) for n=0..59.

Formula

f(x,t)=exp(x*t)*(x*(1 - 2*exp(x)) - 2*exp(x))/(1 - exp(t))

Example

{0, 2, -4},
{0, -1, 6, -6},
{0, 0, -8, 24, -16},
{0, 2, 0, -60, 120, -60},
{0, 0, 48, 0, -480, 720, -288},
{0, -40, 0,840, 0, -4200, 5040, -1680},
{0, 0, -1920, 0, 13440, 0, -40320, 40320, -11520},
{0, 3024, 0, -60480, 0,211680, 0, -423360, 362880, -90720},
{0, 0, 241920, 0, -1612800, 0, 3386880, 0, -4838400, 3628800, -806400},
{0, -604800, 0, 11975040, 0, -39916800, 0, 55883520, 0, -59875200,39916800, -7983360},
{0, 0, -72576000, 0, 479001600, 0, -958003200,0, 958003200, 0, -798336000, 479001600, -87091200}

Mathematica

p[t_] = Exp[x*t](x*(1 - 2*Exp[x]) - 2*Exp[x])/(1 - Exp[t]);
(* Exp part separated as Imaginary by a substitution*)
a = Table[ Re[CoefficientList[2*n!*(n + 2)!*SeriesCoefficient[
      Series[p[t], {t, 0, 30}] /. Exp[x] -> I, n], x]], {n, 0, 10}];
Flatten[a]

Crossrefs

Cf. A048998, A176295.

Sequence in context: A226031 A308460 A244116 * A302937 A363367 A005657

Adjacent sequences: A138130 A138131 A138132 * A138134 A138135 A138136

Keyword

sign,tabf

Author

Roger L. Bagula, Dec 07 2010

Extensions

Edited by N. J. A. Sloane, Jan 01 2011

Status

approved