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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

Table of n, a(n) for n=1..57.

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.)