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A223172 Triangle S(n,k) by rows: coefficients of 6^((n-1)/2)*(x^(1/6)*d/dx)^n when n is odd, and of 6^(n/2)*(x^(5/6)*d/dx)^n when n is even.

%I A223172 #37 Dec 25 2023 18:02:50
%S A223172 1,1,6,7,6,7,84,36,91,156,36,91,1638,1404,216,1729,4446,2052,216,1729,
%T A223172 41496,53352,16416,1296,43225,148200,102600,21600,1296,43225,1296750,
%U A223172 2223000,1026000,162000,7776,1339975,5742750,5301000,1674000,200880,7776
%N A223172 Triangle S(n,k) by rows: coefficients of 6^((n-1)/2)*(x^(1/6)*d/dx)^n when n is odd, and of 6^(n/2)*(x^(5/6)*d/dx)^n when n is even.
%H A223172 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 A223172 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 A223172 Triangle begins:
%e A223172         1;
%e A223172         1,        6;
%e A223172         7,        6;
%e A223172         7,       84,        36;
%e A223172        91,      156,        36;
%e A223172        91,     1638,      1404,      216;
%e A223172      1729,     4446,      2052,      216;
%e A223172      1729,    41496,     53352,    16416,     1296;
%e A223172     43225,   148200,    102600,    21600,     1296;
%e A223172     43225,  1296750,   2223000,  1026000,   162000,    7776;
%e A223172   1339975,  5742750,   5301000,  1674000,   200880,    7776;
%e A223172   1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656;
%p A223172 a[0]:= f(x):
%p A223172 for i from 1 to 13 do
%p A223172 a[i] := simplify(6^((i+1)mod 2)*x^((4((i+1)mod 2)+1)/6)*(diff(a[i-1],x$1 )));
%p A223172 end do;
%Y A223172 Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.
%K A223172 nonn,tabf
%O A223172 0,3
%A A223172 _Udita Katugampola_, Mar 20 2013