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A223170 Triangle S(n,k) by rows: coefficients of 4^((n-1)/2)*(x^(1/4)*d/dx)^n when n is odd, and of 4^(n/2)*(x^(3/4)*d/dx)^n when n is even.

%I A223170 #29 Dec 25 2023 18:02:08
%S A223170 1,1,4,5,4,5,40,16,45,72,16,45,540,432,64,585,1404,624,64,585,9360,
%T A223170 11232,3328,256,9945,31824,21216,4352,256,9945,198900,318240,141440,
%U A223170 21760,1024,208845,835380,742560,228480,26880,1024,208845,5012280,10024560,5940480,1370880,129024,4096
%N A223170 Triangle S(n,k) by rows: coefficients of 4^((n-1)/2)*(x^(1/4)*d/dx)^n when n is odd, and of 4^(n/2)*(x^(3/4)*d/dx)^n when n is even.
%H A223170 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 A223170 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 A223170 Triangle begins:
%e A223170 1;
%e A223170 1, 4;
%e A223170 5, 4;
%e A223170 5, 40, 16;
%e A223170 45, 72, 16;
%e A223170 45, 540, 432, 64;
%e A223170 585, 1404, 624, 64;
%e A223170 585, 9360, 11232, 3328, 256;
%e A223170 9945, 31824, 21216, 4352, 256;
%e A223170 9945, 198900, 318240, 141440, 21760, 1024;
%e A223170 208845, 835380, 742560, 228480, 26880, 1024;
%e A223170 208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096;
%p A223170 a[0]:= f(x):
%p A223170 for i from 1 to 13 do
%p A223170 a[i] := simplify(4^((i+1)mod 2)*x^((2((i+1)mod 2)+1)/4)*(diff(a[i-1],x$1 )));
%p A223170 end do;
%t A223170 nmax = 12;
%t A223170 b[0] = Exp[x]; For[ i = 1 , i <= nmax , i++, b[i] = 4^Mod[i + 1, 2]*x^((2 Mod[i + 1, 2] + 1)/4)*D[b[i - 1], x]] // Simplify;
%t A223170 row[1] = {1}; row[n_] := List @@ Expand[b[n]/f[x]] /. x -> 1;
%t A223170 Table[row[n], {n, 1, nmax}] // Flatten (* _Jean-François Alcover_, Feb 22 2019, from Maple *)
%Y A223170 Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.
%K A223170 nonn,tabf
%O A223170 0,3
%A A223170 _Udita Katugampola_, Mar 20 2013
%E A223170 Missing terms inserted by _Jean-François Alcover_, Feb 22 2019