A138169 Irregular triangle from the expansion of p(x,t) = exp(x*t)/(x - t/2 - t/(exp(t) - 1)).

1, 0, -2, 2, -1, 1, 6, -12, 6, 0, 12, -24, -12, 72, -72, 24, 24, -52, -88, 356, -240, -360, 720, -480, 120, 0, -720, 2280, -1320, -3720, 6360, -1200, -6000, 7200, -3600, 720, -3060, 10260, 2580, -56340, 86760, -12480, -95760, 93240, 12600, -88200, 75600, -30240, 5040

Offset

0,3

References

Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 1965 page 273.

A. Messiah, Quantum Mechanics, vol. 2, p. 712, North Holland, 1969.

Links

G. C. Greubel, Rows n = 0..30 of the irregular triangle, flattened

Formula

Define Sum_{n >= 0} p(x, n) * t^n/n! = exp(x*t)/(x - t/2 - t/(exp(t) - 1)) then T(n, k) = coefficients of ( (n+1)!*(x-1)^(n+1) )*( n!*p(x, n) ).

Sum_{k=0..2*n+1} T(n, k) = 0^n = A000007(n). - G. C. Greubel, Apr 01 2021

Example

Irregular triangle begins as:
   1;
   0,   -2,    2;
  -1,    1,    6,   -12,     6;
   0,   12,  -24,   -12,    72,  -72,    24;
  24,  -52,  -88,   356,  -240, -360,   720,  -480,  120;
   0, -720, 2280, -1320, -3720, 6360, -1200, -6000, 7200, -3600, 720;

Mathematica

Table[CoefficientList[(n+1)!*n!*SeriesCoefficient[Series[(x-1)^(n+1)*Exp[x*t]/(x - t*Coth[t/2]/2), {t, 0, 30}], n], x], {n, 0, 10}]//Flatten (* modified by G. C. Greubel, Apr 01 2021 *)

Crossrefs

Sequence in context: A135879 A176224 A174640 * A139331 A173886 A090441

Adjacent sequences: A138166 A138167 A138168 * A138170 A138171 A138172

Keyword

tabf,sign

Author

Roger L. Bagula, May 04 2008

Extensions

Edited by G. C. Greubel, Apr 01 2021

Status

approved