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 A137338 Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), ordered by rising powers. 3
 1, -1, 1, 0, -3, 1, 3, 6, -6, 1, -12, -9, 26, -10, 1, 45, 3, -109, 71, -15, 1, -198, 81, 501, -475, 155, -21, 1, 1071, -786, -2663, 3329, -1455, 295, -28, 1, -6984, 6711, 16510, -25495, 13729, -3647, 511, -36, 1, 53217, -60309, -117912, 216004, -135961, 43897, -7994, 826, -45, 1, -462330, 589197, 953711 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums are 1, 0, -2, 4, -4, -4, 44, -236, 1300, -8276, 61484, etc. Matrix inverse is A216916. - Peter Luschny, Sep 21 2012 LINKS Carl V. L. Charlier, Über die Darstellung willkürlicher Funktionen, Arkiv För Matematik, Astronomi Och Fysik, Band 2, No. 20 (Meddelande från Lunds Astronomiska Observatorium, Series I, No. 27), 1905, 1-35. [Accessible only in the USA via the HathiTrust Digital Library.] M. Dunster, Uniform asymptotic expansions for Charlier polynomials, J. Approx. Theory, 112 (2001), pp. 93-133. Wikipedia, Carl Charlier. Wikipedia, Charlier polynomials. FORMULA Charlier polynomials: C_{n}(a; x) = Sum_{k=0..n} binomial(n,k)*binomial(x,k)*k!*(-a)^(n-k). EXAMPLE [0]     1, [1]    -1,      1, [2]     0,     -3,       1, [3]     3,      6,      -6,      1, [4]   -12,     -9,      26,    -10,       1, [5]    45,      3,    -109,     71,     -15,     1, [6]  -198,     81,     501,   -475,     155,   -21,     1, [7]  1071,   -786,   -2663,   3329,   -1455,   295,   -28,   1, [8] -6984,   6711,   16510, -25495,   13729, -3647,   511, -36,   1, [9] 53217, -60309, -117912, 216004, -135961, 43897, -7994, 826, -45, 1. MAPLE with(PolynomialTools): C := (n, x) -> if n>0 then expand((x-n)*C(n-1, x)-n*C(n-2, x)) elif n = 0 then 1 else 0 fi: A137338_row := n -> CoefficientList(C(n, x), x); for n from 0 to 7 do A137338_row(n) od; # Peter Luschny, Sep 21 2012 MATHEMATICA Ca[x, -1] = 0; Ca[x, 0] = 1; Ca[x_, n_] := Ca[x, n] = (x - (n - 1) - 1)*Ca[x, n - 1] - n*Ca[x, n - 2]; Table[ExpandAll[Ca[x, n]], {n, 0, 10}]; a = Table[CoefficientList[Ca[x, n], x], {n, 0, 10}]; Flatten[a] CROSSREFS Cf. A216916. Sequence in context: A033789 A109532 A264584 * A176106 A302867 A058659 Adjacent sequences:  A137335 A137336 A137337 * A137339 A137340 A137341 KEYWORD tabl,sign AUTHOR Roger L. Bagula, Apr 07 2008 EXTENSIONS Edited by Peter Luschny, Sep 21 2012 STATUS approved

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Last modified July 24 01:41 EDT 2021. Contains 346269 sequences. (Running on oeis4.)