This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A176230 #65 Jan 29 2022 09:40:54
%S A176230 1,1,1,3,6,1,15,45,15,1,105,420,210,28,1,945,4725,3150,630,45,1,10395,
%T A176230 62370,51975,13860,1485,66,1,135135,945945,945945,315315,45045,3003,
%U A176230 91,1,2027025,16216200,18918900,7567560,1351350,120120,5460,120,1,34459425
%N A176230 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].
%C A176230 Row sums are A066223. Reverse of A119743. Inverse is alternating sign version.
%C A176230 Diagonal sums are essentially A025164.
%C A176230 From _Tom Copeland_, Dec 13 2015: (Start)
%C A176230 See A099174 for relations to the Hermite polynomials and the link for operator relations, including the infinitesimal generator containing A000384.
%C A176230 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.
%C A176230 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.
%C A176230 Divided along the diagonals by the initial element (A001147) of the diagonal, this matrix becomes the even rows of A034839.
%C A176230 (End)
%C A176230 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
%H A176230 G. C. Greubel, <a href="/A176230/b176230.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A176230 Peter Bala, <a href="/A035342/a035342_Bala.txt">Generalized Dobinski formulas</a>
%H A176230 H. Chan, S. Cooper, and P. Toh, <a href="http://dx.doi.org/10.1016/j.aim.2005.12.003">The 26th power of Dedekind's eta function</a> Advances in Mathematics, 207 (2006) 532-543.
%H A176230 Tom Copeland, <a href="http://tcjpn.wordpress.com/2012/11/29/infinigens-the-pascal-pyramid-and-the-witt-and-virasoro-algebras/">Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras</a>, 2012.
%H A176230 Tom Copeland, <a href="https://tcjpn.wordpress.com/2020/07/15/juggling-zeros-in-the-matrix-example-ii/">Juggling Zeros in the Matrix (Example II)</a>, 2020.
%F A176230 Number triangle T(n,k) = (2n)!/((2k)!(n-k)!2^(n-k)).
%F A176230 T(n,k) = A122848(2n,k+n). - _R. J. Mathar_, Jan 14 2011
%F A176230 [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
%F A176230 E.g.f.: exp[t*x/(1-2x)]/(1-2x)^(1/2). - _Tom Copeland_ , Dec 10 2013
%F A176230 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
%F A176230 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
%F A176230 T(n, k) = (-2)^(n-k)*[x^k] KummerU(-n, 1/2, x). - _Peter Luschny_, Jan 18 2020
%e A176230 Triangle begins
%e A176230 1,
%e A176230 1, 1,
%e A176230 3, 6, 1,
%e A176230 15, 45, 15, 1,
%e A176230 105, 420, 210, 28, 1,
%e A176230 945, 4725, 3150, 630, 45, 1,
%e A176230 10395, 62370, 51975, 13860, 1485, 66, 1,
%e A176230 135135, 945945, 945945, 315315, 45045, 3003, 91, 1,
%e A176230 2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1
%e A176230 Production matrix is
%e A176230 1, 1,
%e A176230 2, 5, 1,
%e A176230 0, 12, 9, 1,
%e A176230 0, 0, 30, 13, 1,
%e A176230 0, 0, 0, 56, 17, 1,
%e A176230 0, 0, 0, 0, 90, 21, 1,
%e A176230 0, 0, 0, 0, 0, 132, 25, 1,
%e A176230 0, 0, 0, 0, 0, 0, 182, 29, 1,
%e A176230 0, 0, 0, 0, 0, 0, 0, 240, 33, 1.
%p A176230 ser := n -> series(KummerU(-n, 1/2, x), x, n+1):
%p A176230 seq(seq((-2)^(n-k)*coeff(ser(n), x, k), k=0..n), n=0..8); # _Peter Luschny_, Jan 18 2020
%t A176230 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 *)
%Y A176230 Cf. A113278.
%Y A176230 Cf. A000384, A001147, A034839, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299.
%K A176230 easy,nonn,tabl
%O A176230 0,4
%A A176230 _Paul Barry_, Apr 12 2010