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 A137346 Coefficients of a special case of Poisson-Charlier polynomials. a0=2; G.f.: Exp[ -a0*t]*(1 + t)^x; Ca(x, n) = (x - (n - 1) - 2)*Ca(x, n - 1) - 2*(n - 1)*Ca(x, n - 2). 0
 1, -2, 1, 4, -5, 1, -8, 20, -9, 1, 16, -78, 59, -14, 1, -32, 324, -360, 135, -20, 1, 64, -1520, 2254, -1165, 265, -27, 1, -128, 8336, -15232, 9954, -3045, 469, -35, 1, 256, -53872, 113868, -88508, 33649, -6888, 770, -44, 1, -512, 405600, -948840, 839684, -376278, 95025, -14028, 1194, -54, 1, 1024, -3492416 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row sums are {1, -1, 0, 4, -16, 48, -128, 320, -768, 1792, -4096}. The signs from this recursion are different from those in A046716. REFERENCES M. Dunster, Uniform asymptotic expansions for Charlier polynomials, J. Approx. Theory, 112 (2001) pp. 93-133, http://www.rohan.sdsu.edu/~dunster/Charlier.pdf LINKS FORMULA a0=2; G.f.: Exp[ -a0*t]*(1 + t)^x; Ca(x, n) = (x - (n - 1) - 2)*Ca(x, n - 1) - 2*(n - 1)*Ca(x, n - 2); EXAMPLE {1}, {-2, 1}, {4, -5, 1}, {-8, 20, -9, 1}, {16, -78,59, -14, 1}, {-32, 324, -360, 135, -20, 1}, {64, -1520, 2254, -1165, 265, -27, 1}, {-128, 8336, -15232, 9954, -3045, 469, -35, 1}, {256, -53872, 113868, -88508, 33649, -6888, 770, -44, 1}, {-512, 405600, -948840, 839684, -376278, 95025, -14028, 1194, -54, 1}, {1024, -3492416, 8793216, -8592220,4373060, -1297569, 235473, -26370, 1770, -65, 1} MATHEMATICA (*Coefficients from the expansion*) Clear[p, a, a0] a0 = 2; p[t_] = Exp[ -a0*t]*(1 + t)^x; g = Table[ ExpandAll[n!SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[ExpandAll[n!SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], x], {n, 0, 10}]; Flatten[a] (* coefficients from the polynomial recursion*) Clear[Ca] Ca[x, -1] = 0; Ca[x, 0] = 1; Ca[x, 1] = -2 + x; Ca[x_, n_] := Ca[x, n] = (x - (n - 1) - 2)*Ca[x, n - 1] - 2*(n - 1)*Ca[x, n - 2]; Table[ExpandAll[Ca[x, n]], {n, 0, 10}]; a2 = Table[CoefficientList[Ca[x, n], x], {n, 0, 10}] Flatten[a2] CROSSREFS Cf. A046716. Sequence in context: A154342 A143494 A124960 * A264017 A159971 A114158 Adjacent sequences:  A137343 A137344 A137345 * A137347 A137348 A137349 KEYWORD uned,tabl,sign AUTHOR Roger L. Bagula, Apr 08 2008 STATUS approved

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Last modified April 26 12:00 EDT 2019. Contains 322472 sequences. (Running on oeis4.)