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A176231 Coefficient array of orthogonal polynomials whose moment sequence is the double factorial numbers A001147.

%I A176231 #14 Jan 15 2020 10:23:38
%S A176231 1,-1,1,3,-6,1,-15,45,-15,1,105,-420,210,-28,1,-945,4725,-3150,630,
%T A176231 -45,1,10395,-62370,51975,-13860,1485,-66,1,-135135,945945,-945945,
%U A176231 315315,-45045,3003,-91,1,2027025,-16216200,18918900,-7567560,1351350,-120120,5460,-120,1
%N A176231 Coefficient array of orthogonal polynomials whose moment sequence is the double factorial numbers A001147.
%C A176231 Exponential Riordan array [1/sqrt(1+2x),x/(1+2x)]. Inverse of A176230.
%C A176231 Diagonal sums are an alternating sign version of A025164.
%F A176231 Number triangle T(n,k) = (-1)^(n-k)*(2n)!/((2k)!(n-k)!2^(n-k)).
%F A176231 He_(2*n)(x) = Sum_{k=0..n} T(n, k)*x^(2*k) where He is Hermite's polynomial. - _Michael Somos_, Jan 15 2020
%e A176231 Triangle begins
%e A176231   1,
%e A176231   -1, 1,
%e A176231   3, -6, 1,
%e A176231   -15, 45, -15, 1,
%e A176231   105, -420, 210, -28, 1,
%e A176231   -945, 4725, -3150, 630, -45, 1,
%e A176231   10395, -62370, 51975, -13860, 1485, -66, 1,
%e A176231   -135135, 945945, -945945, 315315, -45045, 3003, -91, 1,
%e A176231   2027025, -16216200, 18918900, -7567560, 1351350, -120120, 5460, -120, 1
%e A176231 Production matrix is
%e A176231   -1, 1,
%e A176231   2, -5, 1,
%e A176231   0, 12, -9, 1,
%e A176231   0, 0, 30, -13, 1,
%e A176231   0, 0, 0, 56, -17, 1,
%e A176231   0, 0, 0, 0, 90, -21, 1,
%e A176231   0, 0, 0, 0, 0, 132, -25, 1,
%e A176231   0, 0, 0, 0, 0, 0, 182, -29, 1,
%e A176231   0, 0, 0, 0, 0, 0, 0, 240, -33, 1
%p A176231 T := (n,k) -> (2*n)!*(-1/2)^(n-k)/(2*k)!*(n-k)!:
%p A176231 seq(seq(T(n,k), k=0..n), n=0..8); # _Peter Luschny_, Jul 20 2019
%t A176231 (* The function RiordanArray is defined in A256893. *)
%t A176231 rows = 9;
%t A176231 R = RiordanArray[1/Sqrt[1 + 2 #]&, #/(1 + 2 #)&, rows, True];
%t A176231 R // Flatten (* _Jean-François Alcover_, Jul 20 2019 *)
%t A176231 T[ n_, k_] := Coefficient[ HermiteH[2 n, x/Sqrt[2]], x, 2 k]/2^n; (* _Michael Somos_, Jan 15 2020 *)
%t A176231 T[ n_, k_] := Coefficient[ Nest[# x - D[#, x]&, 1, 2 n], x, 2 k]; (* _Michael Somos_, Jan 15 2020 *)
%o A176231 (PARI) {T(n, k) = my(t=1); for(i=1, 2*n, t = x*t - t'); polcoeff(t, 2*k)}; /* _Michael Somos_, Jan 15 2020 */
%Y A176231 Cf. A001147, A025164, A176230.
%K A176231 easy,sign,tabl
%O A176231 0,4
%A A176231 _Paul Barry_, Apr 12 2010