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A137784
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Triangular sequence of coefficients of expansion of a degenerate partition of the Bernoulli generating function and the Hermite generating function: 1) b(x,t)=t*Exp[t*x)/(Exp[t]-1); 2) h(x,t)=Exp[2*x*t-t^2]; to give p(x,t)=t*Ex[3*x*t-t^2)/(Exp[t]-1).
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0
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2, -3, 18, -44, -72, 216, 360, -1980, -1620, 3240, 7176, 25920, -71280, -38880, 58320, -151200, 753480, 1360800, -2494800, -1020600, 1224720, -3587520, -21772800, 54250560, 65318400, -89812800, -29393280, 29393280, 152409600, -678041280, -2057529600, 3417785280, 3086294400
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Row sums are:
{2, 15, 100, 0, -18744, -327600,4395840, 393724800, 4044781440, -611804793600, -28321891315200};
The partition of two orthogonal Hilbert spaces is used to get wave functions for two dimensional systems in quantum mechanics.
Here the result is that x is space like and t is time like in a two dimensional vibrator system.
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REFERENCES
| Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 87-100
Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 1965 pp 282-290
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FORMULA
| p(x,t)=t*Ex[3*x*t-t^2)/(Exp[t]-1)=Sum(P(x,n)*t^n/n!,{n,0,Infinity}); out-n,m=(n+2)!(n!*Coefficients(Q(x,n)).
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EXAMPLE
| {2},
{-3, 18},
{-44, -72, 216},
{360, -1980, -1620, 3240},
{7176, 25920, -71280, -38880,58320},
{-151200, 753480, 1360800, -2494800, -1020600, 1224720},
{-3587520, -21772800, 54250560, 65318400, -89812800, -29393280, 29393280},
{152409600, -678041280, -2057529600, 3417785280, 3086294400, -3394923840,-925888320, 793618560},
{3957690240, 36578304000, -81364953600, -164602368000, 205067116800, 148142131200, -135796953600, -31744742400, 23808556800},
{-301771008000, 1175434001280, 5431878144000, -8055130406400, -12221725824000, 12180986737920, 7333035494400, -5761670745600, -1178523561600, 785682374400}, {-8017013491200, -108637562880000, 211578120230400, 651825377280000, -724961736576000, -879964259328000, 730859204275200, 377127539712000, -259275183552000, -47140942464000, 28284565478400}
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MATHEMATICA
| Clear[p, b, a] p[t_] = FullSimplify[(t*Exp[x*t]/(Exp[t] - 1))*Exp[2*x*t - t^2]]; Table[ ExpandAll[(n + 2)!*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[(n + 2)!*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]
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CROSSREFS
| Sequence in context: A134850 A163910 A064777 * A053195 A003693 A048047
Adjacent sequences: A137781 A137782 A137783 * A137785 A137786 A137787
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KEYWORD
| tabl,uned,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 28 2008
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