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A138169
Irregular triangle from the expansion of p(x,t) = exp(x*t)/(x - t/2 - t/(exp(t) - 1)).
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.
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
KEYWORD
tabf,sign
AUTHOR
Roger L. Bagula, May 04 2008
EXTENSIONS
Edited by G. C. Greubel, Apr 01 2021
STATUS
approved