%I #73 Dec 25 2023 18:01:25
%S 1,1,2,3,2,3,12,4,15,20,4,15,90,60,8,105,210,84,8,105,840,840,224,16,
%T 945,2520,1512,288,16,945,9450,12600,5040,720,32,10395,34650,27720,
%U 7920,880,32,10395,124740,207900,110880,23760,2112,64,135135,540540,540540,205920,34320,2496,64
%N Triangle S(n, k) by rows: coefficients of 2^((n-1)/2)*(x^(1/2)*d/dx)^n when n is odd, and of 2^(n/2)*(x^(1/2)*d/dx)^n when n is even.
%C Also coefficients in the expansion of k-th derivative of exp(n*x^2), see Mathematica program. - _Vaclav Kotesovec_, Jul 16 2013
%H Vincenzo Librandi, <a href="/A223168/b223168.txt">Rows n = 0..60, flattened</a>
%H U. N. Katugampola, <a href="http://dx.doi.org/10.1016/j.amc.2014.12.067">Mellin Transforms of Generalized Fractional Integrals and Derivatives</a>, Appl. Math. Comput. 257(2015) 566-580.
%H U. N. Katugampola, <a href="http://arxiv.org/abs/1411.5229">Existence and Uniqueness results for a class of Generalized Fractional Differential Equations</a>, arXiv preprint arXiv:1411.5229 [math.CA], 2014.
%e Triangle begins:
%e 1;
%e 1, 2;
%e 3, 2;
%e 3, 12, 4;
%e 15, 20, 4;
%e 15, 90, 60, 8;
%e 105, 210, 84, 8;
%e 105, 840, 840, 224, 16;
%e 945, 2520, 1512, 288, 16;
%e 945, 9450, 12600, 5040, 720, 32;
%e 10395, 34650, 27720, 7920, 880, 32;
%e 10395, 124740, 207900, 110880, 23760, 2112, 64;
%e 135135, 540540, 540540, 205920, 34320, 2496, 64;
%e .
%e Expansion takes the form:
%e 2^0 (x^(1/2)*d/dx)^1 = 1*x^(1/2)*d/dx.
%e 2^1 (x^(1/2)*d/dx)^2 = 1*d/dx + 2*x*d^2/dx^2.
%e 2^1 (x^(1/2)*d/dx)^3 = 3*x^(1/2)*d^2/dx^2 + 2*x^(3/2)*d^3/dx^3.
%e 2^2 (x^(1/2)*d/dx)^4 = 3*d^2/dx^2 + 12*x*d^3/dx^3 + 4*x^2*d^4/dx^4.
%e 2^2 (x^(1/2)*d/dx)^5 = 15*x^(1/2)*d^3/dx^3 + 20*x^(3/2)*d^4/dx^4 + 4*x^(5/2)*d^5/dx^5.
%e `
%e `
%p a[0]:= f(x);
%p for i from 1 to 13 do
%p a[i]:= simplify(2^((i+1)mod 2)*x^(1/2)*(diff(a[i-1],x$1)));
%p end do;
%t Flatten[CoefficientList[Expand[FullSimplify[Table[D[E^(n*x^2),{x,k}]/(E^(n*x^2)*(2*n)^Floor[(k+1)/2]),{k,1,13}]]]/.x->1,n]] (* _Vaclav Kotesovec_, Jul 16 2013 *)
%Y Odd rows includes absolute values of A098503 from right to left.
%Y Cf. A223169-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.
%K nonn,tabf
%O 0,3
%A _Udita Katugampola_, Mar 17 2013