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

%I A223524 #37 Mar 30 2025 09:51:57
%S A223524 1,1,2,3,12,4,15,90,60,8,105,840,840,224,16,945,9450,12600,5040,720,
%T A223524 32,10395,124740,207900,110880,23760,2112,64,135135,1891890,3783780,
%U A223524 2522520,720720,96096,5824,128,2027025,32432400
%N A223524 Triangle S(n, k) by rows: coefficients of 2^(n/2)*(x^(1/2)*d/dx)^n, where n =0, 2, 4, 6, ...
%C A223524 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
%H A223524 Udita N. Katugampola, <a href="https://arxiv.org/abs/1112.6031">Mellin Transforms of Generalized Fractional Integrals and Derivatives</a>, arXiv:1112.6031 [math.CA], 2011-2014; Appl. Math. Comput. 257(2015) 566-580.
%H A223524 Udita 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 [math.CA], 2014-2016.
%e A223524 Triangle begins:
%e A223524   1;
%e A223524   1, 2;
%e A223524   3, 12, 4;
%e A223524   15, 90, 60, 8;
%e A223524   105, 840, 840, 224, 16;
%e A223524   945, 9450, 12600, 5040, 720, 32;
%e A223524   10395, 124740, 207900, 110880, 23760, 2112, 64;
%e A223524   ...
%e A223524 Expansion takes the form:
%e A223524   2^1 (x^(1/2)*d/dx)^2 = 1*d/dx + 2*x*d^2/dx^2.
%e A223524   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.
%p A223524 a[0]:= f(x):
%p A223524 for i from 1 to 20 do
%p A223524 a[i]:= simplify(2^((i+1)mod 2)*x^(1/2)*(diff(a[i-1],x$1)));
%p A223524 end do:
%p A223524 for j from 1 to 10 do
%p A223524 b[j]:=a[2j];
%p A223524 end do;
%t A223524 Flatten[Abs[Table[CoefficientList[2^n n! LaguerreL[n, -1/2, x], x], {n, 0, 7}]]] (* _Ali Pourzand_, Mar 28 2025 *)
%Y A223524 Rows includes even rows of A223168.
%Y A223524 Cf. A223168-A223172 and A223523.
%K A223524 nonn,tabl
%O A223524 1,3
%A A223524 _Udita Katugampola_, Mar 21 2013