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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. 2
1, 1, 4, 5, 4, 5, 40, 16, 45, 72, 16, 45, 540, 432, 64, 585, 1404, 624, 64, 585, 9360, 11232, 3328, 256, 9945, 31824, 21216, 4352, 256, 9945, 198900, 318240, 141440, 21760, 1024, 208845, 835380, 742560, 228480, 26880, 1024, 208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..47.

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, 2014

EXAMPLE

Triangle begins:

1;

1, 4;

5, 4;

5, 40, 16;

45, 72, 16;

45, 540, 432, 64;

585, 1404, 624, 64;

585, 9360, 11232, 3328, 256;

9945, 31824, 21216, 4352, 256;

9945, 198900, 318240, 141440, 21760, 1024;

208845, 835380, 742560, 228480, 26880, 1024;

208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096;

MAPLE

a[0]:= f(x):

for i from 1 to 13 do

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

end do;

MATHEMATICA

nmax = 12;

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;

row[1] = {1}; row[n_] := List @@ Expand[b[n]/f[x]] /. x -> 1;

Table[row[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Feb 22 2019, from Maple *)

CROSSREFS

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

Sequence in context: A276868 A182495 A265300 * A303275 A198817 A248624

Adjacent sequences:  A223167 A223168 A223169 * A223171 A223172 A223173

KEYWORD

nonn,tabf

AUTHOR

Udita Katugampola, Mar 20 2013

EXTENSIONS

Missing terms inserted by Jean-François Alcover, Feb 22 2019

STATUS

approved

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Last modified July 31 05:53 EDT 2021. Contains 346367 sequences. (Running on oeis4.)