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.

1, 1, 6, 7, 6, 7, 84, 36, 91, 156, 36, 91, 1638, 1404, 216, 1729, 4446, 2052, 216, 1729, 41496, 53352, 16416, 1296, 43225, 148200, 102600, 21600, 1296, 43225, 1296750, 2223000, 1026000, 162000, 7776, 1339975, 5742750, 5301000, 1674000, 200880, 7776

Offset: 0

Author

Udita Katugampola, Mar 20 2013

Keywords

• nonn
• tabf

Examples

			Triangle begins:
        1;
        1,        6;
        7,        6;
        7,       84,        36;
       91,      156,        36;
       91,     1638,      1404,      216;
     1729,     4446,      2052,      216;
     1729,    41496,     53352,    16416,     1296;
    43225,   148200,    102600,    21600,     1296;
    43225,  1296750,   2223000,  1026000,   162000,    7776;
  1339975,  5742750,   5301000,  1674000,   200880,    7776;
  1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656;
		

Links

• U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
• U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229 [math.CA], 2014.

Crossrefs

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

Programs

Maple

a[0]:= f(x):
for i from 1 to 13 do
a[i] := simplify(6^((i+1)mod 2)*x^((4((i+1)mod 2)+1)/6)*(diff(a[i-1],x$1 )));
end do;