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A137346 Coefficients of a special case of Poisson-Charlier polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.

%I A137346 #39 Feb 16 2025 08:33:07
%S A137346 1,-2,1,4,-5,1,-8,20,-9,1,16,-78,59,-14,1,-32,324,-360,135,-20,1,64,
%T A137346 -1520,2254,-1165,265,-27,1,-128,8336,-15232,9954,-3045,469,-35,1,256,
%U A137346 -53872,113868,-88508,33649,-6888,770,-44,1,-512,405600,-948840,839684,-376278,95025,-14028,1194,-54
%N A137346 Coefficients of a special case of Poisson-Charlier polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.
%H A137346 M. Dunster, <a href="https://doi.org/10.1006/jath.2001.3595">Uniform asymptotic expansions for Charlier polynomials</a>, J. Approx. Theory, 112 (2001) pp. 93-133.
%H A137346 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Poisson-CharlierPolynomial.html">Poisson-Charlier Polynomial</a>.
%F A137346 T(n, k) = n!*[x^k] p(n) where p(n) = [t^n] exp(-2*t)*(1+t)^x.
%F A137346 With p(0, x) = 1 and p(1, x) = x - 2 the polynomials obey the recurrence
%F A137346 p(n, x) = (x - n - 1)*p(n-1, x) - 2*(n - 1)*p(n-2, x).
%F A137346 Row sums are (-2)^n*(n-1) = (-1)^n*A159964(n-1).
%F A137346 From _Peter Bala_, Oct 23 2019: (Start)
%F A137346 The unsigned row polynomials are
%F A137346    R(n,x) = Sum_{k=0..n} (-1)^k*binomial(n, k)*k!*2^(n-k)*binomial(-x, k).
%F A137346 They occur in series acceleration formulas for the constant
%F A137346    1/e^2 = n!*2^n*Sum_{k >= 0}(-2)^k/(k!*R(n,k)*R(n,k+1)) = 0.1353 35283 23661 ... (cf. A092553, A046716, A094816).
%F A137346 (End)
%F A137346 R(n, x) = KummerU(-n, 1 - n - x, 2). - _Peter Luschny_, Oct 27 2019
%e A137346 {1},
%e A137346 {-2, 1},
%e A137346 {4, -5, 1},
%e A137346 {-8, 20, -9, 1},
%e A137346 {16, -78,59, -14, 1},
%e A137346 {-32, 324, -360, 135, -20, 1},
%e A137346 {64, -1520, 2254, -1165, 265, -27, 1},
%e A137346 {-128, 8336, -15232, 9954, -3045, 469, -35, 1},
%e A137346 {256, -53872, 113868, -88508, 33649, -6888, 770, -44, 1},
%e A137346 {-512, 405600, -948840, 839684, -376278, 95025, -14028, 1194, -54, 1},
%e A137346 {1024, -3492416, 8793216, -8592220,4373060, -1297569, 235473, -26370, 1770, -65, 1}
%p A137346 R := proc(n) add((-1)^k*binomial(n,k)* k!*2^(n-k)*binomial(-x, k), k=0..n);
%p A137346 expand(%) end: p := n -> seq((-1)^(n-k)*coeff(R(n), x, k), k=0..n):
%p A137346 seq(p(n), n = 0..9);
%p A137346 # Or:
%p A137346 egf := exp(-2*t)*(1+t)^x: ser := series(egf, t, 12): p := n -> coeff(ser, t, n):
%p A137346 seq(n!*seq(coeff(p(n), x, k), k=0..n), n=0..9); # _Peter Luschny_, Oct 27 2019
%t A137346 Ca[x, 0] = 1; Ca[x, 1] = -2 + x;
%t A137346 Ca[x_, n_] := Ca[x, n] = (x - n - 1) Ca[x, n - 1] - 2 (n - 1) Ca[x, n - 2];
%t A137346 Table[CoefficientList[Ca[x, n], x], {n, 0, 9}] // Flatten
%t A137346 (* The unsigned row polynomials (see _Peter Bala_'s comment) are: *)
%t A137346 R[n_] := HypergeometricU[-n, 1 - n - x, 2];
%t A137346 Table[R[n], {n, 0, 6}] (* _Peter Luschny_, Oct 27 2019 *)
%Y A137346 Cf. A046716, A092553, A094816, A159964.
%K A137346 tabl,sign
%O A137346 0,2
%A A137346 _Roger L. Bagula_, Apr 08 2008
%E A137346 Edited by _Peter Luschny_, Oct 27 2019