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A223527 Triangle S(n,k) by rows: coefficients of 4^((n-1)/2)*(x^(1/4)*d/dx)^n when n=1,3,5,...

%I A223527 #24 Jan 10 2024 15:59:31
%S A223527 1,5,4,45,72,16,585,1404,624,64,9945,31824,21216,4352,256,208845,
%T A223527 835380,742560,228480,26880,1024,5221125,25061400,27846000,11424000,
%U A223527 2016000,153600,4096,151412625,847910700,1130547600,579768000,136416000,15590400,831488,16384
%N A223527 Triangle S(n,k) by rows: coefficients of 4^((n-1)/2)*(x^(1/4)*d/dx)^n when n=1,3,5,...
%H A223527 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 A223527 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 A223527 Triangle begins:
%e A223527 1;
%e A223527 5, 4;
%e A223527 45, 72, 16;
%e A223527 585, 1404, 624, 64;
%e A223527 9945, 31824, 21216, 4352, 256;
%e A223527 208845, 835380, 742560, 228480, 26880, 1024;
%e A223527 5221125, 25061400, 27846000, 11424000, 2016000, 153600, 4096;
%e A223527 151412625, 847910700, 1130547600, 579768000, 136416000, 15590400, 831488, 16384;
%p A223527 a[0]:= f(x):
%p A223527 for i from 1 to 20 do
%p A223527 a[i] := simplify(4^((i+1)mod 2)*x^((2((i+1)mod 2)+1)/4)*(diff(a[i-1],x$1 )));
%p A223527 end do:
%p A223527 for j from 1 to 10 do
%p A223527 b[j]:=a[2j-1];
%p A223527 end do;
%Y A223527 Odd rows of A223170.
%Y A223527 Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.
%K A223527 nonn,tabl
%O A223527 1,2
%A A223527 _Udita Katugampola_, Mar 23 2013