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A223169 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)*(x^(1/3)*d/dx)^n when n is odd, and of 3^(n/2)*(x^(2/3)*d/dx)^n when n is even.

%I A223169 #34 Dec 25 2023 18:01:42
%S A223169 1,1,3,4,3,4,24,9,28,42,9,28,252,189,27,280,630,270,27,280,3360,3780,
%T A223169 1080,81,3640,10920,7020,1404,81,3640,54600,81900,35100,5265,243,
%U A223169 58240,218400,187200,56160,6480,243,58240,1048320,1965600
%N A223169 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)*(x^(1/3)*d/dx)^n when n is odd, and of 3^(n/2)*(x^(2/3)*d/dx)^n when n is even.
%H A223169 U. N. Katugampola, <a href="http://authors.elsevier.com/a/1QhUNLvMg0Zs~">Mellin Transforms of Generalized Fractional Integrals and Derivatives</a>, Appl. Math. Comput. 257(2015) 566-580.
%H A223169 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, 2014
%e A223169 Triangle begins:
%e A223169 1;
%e A223169 1, 3;
%e A223169 4, 3;
%e A223169 4, 24, 9;
%e A223169 28, 42, 9;
%e A223169 28, 252, 189, 27;
%e A223169 280, 630, 270, 27;
%e A223169 280, 3360, 3780, 1080, 81;
%e A223169 3640, 10920, 7020, 1404, 81;
%e A223169 3640, 54600, 81900, 35100, 5265, 243,
%e A223169 58240, 218400, 187200, 56160, 6480, 243
%p A223169 a[0]:= f(x):
%p A223169 for i from 1 to 13 do
%p A223169 a[i] := simplify(3^((i+1)mod 2)*x^(((i+1)mod 2+1)/3)*(diff(a[i-1],x$1 )));
%p A223169 end do;
%Y A223169 Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.
%K A223169 nonn,tabf
%O A223169 0,3
%A A223169 _Udita Katugampola_, Mar 18 2013