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A176231
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Coefficient array of orthogonal polynomials whose moment sequence is the double factorial numbers A001147.
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1
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1, -1, 1, 3, -6, 1, -15, 45, -15, 1, 105, -420, 210, -28, 1, -945, 4725, -3150, 630, -45, 1, 10395, -62370, 51975, -13860, 1485, -66, 1, -135135, 945945, -945945, 315315, -45045, 3003, -91, 1, 2027025, -16216200, 18918900, -7567560, 1351350, -120120, 5460, -120, 1
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OFFSET
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0,4
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COMMENTS
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Exponential Riordan array [1/sqrt(1+2x),x/(1+2x)]. Inverse of A176230.
Diagonal sums are an alternating sign version of A025164.
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LINKS
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FORMULA
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Number triangle T(n,k) = (-1)^(n-k)*(2n)!/((2k)!(n-k)!2^(n-k)).
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
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EXAMPLE
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Triangle begins
1,
-1, 1,
3, -6, 1,
-15, 45, -15, 1,
105, -420, 210, -28, 1,
-945, 4725, -3150, 630, -45, 1,
10395, -62370, 51975, -13860, 1485, -66, 1,
-135135, 945945, -945945, 315315, -45045, 3003, -91, 1,
2027025, -16216200, 18918900, -7567560, 1351350, -120120, 5460, -120, 1
Production matrix is
-1, 1,
2, -5, 1,
0, 12, -9, 1,
0, 0, 30, -13, 1,
0, 0, 0, 56, -17, 1,
0, 0, 0, 0, 90, -21, 1,
0, 0, 0, 0, 0, 132, -25, 1,
0, 0, 0, 0, 0, 0, 182, -29, 1,
0, 0, 0, 0, 0, 0, 0, 240, -33, 1
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MAPLE
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T := (n, k) -> (2*n)!*(-1/2)^(n-k)/(2*k)!*(n-k)!:
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MATHEMATICA
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(* The function RiordanArray is defined in A256893. *)
rows = 9;
R = RiordanArray[1/Sqrt[1 + 2 #]&, #/(1 + 2 #)&, rows, True];
T[ n_, k_] := Coefficient[ HermiteH[2 n, x/Sqrt[2]], x, 2 k]/2^n; (* Michael Somos, Jan 15 2020 *)
T[ n_, k_] := Coefficient[ Nest[# x - D[#, x]&, 1, 2 n], x, 2 k]; (* Michael Somos, Jan 15 2020 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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