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Triangle S(n,k) by rows: coefficients of 6^(n/2)*(x^(5/6)*d/dx)^n when n=0,2,4,6,...
26

%I #23 Aug 14 2015 21:16:22

%S 1,1,6,7,84,36,91,1638,1404,216,1729,41496,53352,16416,1296,43225,

%T 1296750,2223000,1026000,162000,7776,1339975,48239100,103369500,

%U 63612000,15066000,1446336,46656,49579075,2082321150,5354540100,4118877000,1300698000,187300512

%N Triangle S(n,k) by rows: coefficients of 6^(n/2)*(x^(5/6)*d/dx)^n when n=0,2,4,6,...

%H 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 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 Triangle begins:

%e 1;

%e 1, 6;

%e 7, 84, 36;

%e 91, 1638, 1404, 216;

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

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

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

%e 49579075, 2082321150, 5354540100, 4118877000, 1300698000, 187300512, 12083904, 279936;

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

%p for i from 1 to 20 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:

%p for j from 1 to 10 do

%p b[j]:=a[2j];

%p end do;

%Y Even rows of A223172.

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

%K nonn,tabl

%O 1,3

%A _Udita Katugampola_, Mar 23 2013