A223524 Triangle S(n, k) by rows: coefficients of 2^(n/2)*(x^(1/2)*d/dx)^n, where n =0, 2, 4, 6, ...

1, 1, 2, 3, 12, 4, 15, 90, 60, 8, 105, 840, 840, 224, 16, 945, 9450, 12600, 5040, 720, 32, 10395, 124740, 207900, 110880, 23760, 2112, 64, 135135, 1891890, 3783780, 2522520, 720720, 96096, 5824, 128, 2027025, 32432400

Offset: 1

Author

Udita Katugampola, Mar 21 2013

Keywords

• nonn
• tabl

Comments

Triangle by rows of the coefficients of 2^n * n! *|L(n,-1/2,x)|, with L the generalized Laguerre polynomials. - Ali Pourzand, Mar 28 2025

Examples

			Triangle begins:
  1;
  1, 2;
  3, 12, 4;
  15, 90, 60, 8;
  105, 840, 840, 224, 16;
  945, 9450, 12600, 5040, 720, 32;
  10395, 124740, 207900, 110880, 23760, 2112, 64;
  ...
Expansion takes the form:
  2^1 (x^(1/2)*d/dx)^2 = 1*d/dx + 2*x*d^2/dx^2.
  2^2 (x^(1/2)*d/dx)^4 = 3*d^2/dx^2 + 12*x*d^3/dx^3 + 4*x^2*d^4/dx^4.
		

Links

• Udita N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, arXiv:1112.6031 [math.CA], 2011-2014; Appl. Math. Comput. 257(2015) 566-580.
• Udita N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229 [math.CA], 2014-2016.

Crossrefs

Rows includes even rows of A223168.

Cf. A223168-A223172 and A223523.

Programs

Maple

a[0]:= f(x):
for i from 1 to 20 do
a[i]:= simplify(2^((i+1)mod 2)*x^(1/2)*(diff(a[i-1],x$1)));
end do:
for j from 1 to 10 do
b[j]:=a[2j];
end do;

Mathematica

Flatten[Abs[Table[CoefficientList[2^n n! LaguerreL[n, -1/2, x], x], {n, 0, 7}]]] (* Ali Pourzand, Mar 28 2025 *)