A223172 - OEIS

%I #37 Dec 25 2023 18:02:50

%S 1,1,6,7,6,7,84,36,91,156,36,91,1638,1404,216,1729,4446,2052,216,1729,

%T 41496,53352,16416,1296,43225,148200,102600,21600,1296,43225,1296750,

%U 2223000,1026000,162000,7776,1339975,5742750,5301000,1674000,200880,7776

%N 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 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,        6;

%e         7,        6;

%e         7,       84,        36;

%e        91,      156,        36;

%e        91,     1638,      1404,      216;

%e      1729,     4446,      2052,      216;

%e      1729,    41496,     53352,    16416,     1296;

%e     43225,   148200,    102600,    21600,     1296;

%e     43225,  1296750,   2223000,  1026000,   162000,    7776;

%e   1339975,  5742750,   5301000,  1674000,   200880,    7776;

%e   1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656;

%p a[0]:= f(x):

%p for i from 1 to 13 do

%p a[i] := simplify(6^((i+1)mod 2)*x^((4((i+1)mod 2)+1)/6)*(diff(a[i-1],x$1 )));

%p end do;

%Y Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

%K nonn,tabf

%O 0,3

%A _Udita Katugampola_, Mar 20 2013