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A176230
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Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].
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6
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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, 34459425
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OFFSET
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0,4
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COMMENTS
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Row sums are A066223. Reverse of A119743. Inverse is alternating sign version.
Diagonal sums are essentially A025164.
See A099174 for relations to the Hermite polynomials and the link for operator relations, including the infinitesimal generator containing A000384.
Row polynomials are 2^n n! Lag(n,-x/2,-1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q.
Divided along the diagonals by the initial element (A001147) of the diagonal, this matrix becomes the even rows of A034839.
(End)
The first few rows appear in expansions related to the Dedekind eta function on pp. 537-538 of the Chan et al. link. - Tom Copeland, Dec 14 2016
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LINKS
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FORMULA
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Number triangle T(n,k) = (2n)!/((2k)!(n-k)!2^(n-k)).
[x^(1/2)(1+2D)]^2 p(n,x)= p(n+1,x) and [D/(1+2D)]p(n,x)= n p(n-1,x) for the row polynomials of T, with D=d/dx. - Tom Copeland, Dec 26 2012
E.g.f.: exp[t*x/(1-2x)]/(1-2x)^(1/2). - Tom Copeland , Dec 10 2013
The n-th row polynomial R(n,x) is given by the type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} (2*k+1)*(2*k+3)*...*(2*k+1+2*(n-1))*(x/2)^k/k!. Cf. A113278. - Peter Bala, Jun 23 2014
The raising operator in my 2012 formula expanded is R = [x^(1/2)(1+2D)]^2 = 1 + x + (2 + 4x) D + 4x D^2, which in matrix form acting on an o.g.f. (formal power series) is the transpose of the production array below. The linear term x is the diagonal of ones after transposition. The main diagonal comes from (1 + 4xD) x^n = (1 + 4n) x^n. The last diagonal comes from (2 D + 4 x D^2) x^n = (2 + 4 xD) D x^n = n * (2 + 4(n-1)) x^(n-1). - Tom Copeland, Dec 13 2015
T(n, k) = (-2)^(n-k)*[x^k] KummerU(-n, 1/2, x). - Peter Luschny, Jan 18 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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ser := n -> series(KummerU(-n, 1/2, x), x, n+1):
seq(seq((-2)^(n-k)*coeff(ser(n), x, k), k=0..n), n=0..8); # Peter Luschny, Jan 18 2020
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MATHEMATICA
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t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); u[n_, k_] := t[2 n, k + n]; Table[ u[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jan 14 2011 *)
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CROSSREFS
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Cf. A000384, A001147, A034839, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299.
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KEYWORD
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AUTHOR
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STATUS
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approved
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