A176230 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].

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

Offset: 0

Author

Paul Barry, Apr 12 2010

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A066223. Reverse of A119743. Inverse is alternating sign version.

Diagonal sums are essentially A025164.

From Tom Copeland, Dec 13 2015: (Start)

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.

The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.

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

Examples

			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.
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Peter Bala, Generalized Dobinski formulas
• H. Chan, S. Cooper, and P. Toh, The 26th power of Dedekind's eta function Advances in Mathematics, 207 (2006) 532-543.
• Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.
• Tom Copeland, Juggling Zeros in the Matrix (Example II), 2020.

Crossrefs

Cf. A113278.

Cf. A000384, A001147, A034839, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299.

Programs

Maple

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

Mathematica

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 *)

Formula

Number triangle T(n,k) = (2n)!/((2k)!(n-k)!2^(n-k)).

T(n,k) = A122848(2n,k+n). - R. J. Mathar, Jan 14 2011

[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