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A223168 Triangle S(n, k) by rows: coefficients of 2^((n-1)/2)*(x^(1/2)*d/dx)^n when n is odd, and of 2^(n/2)*(x^(1/2)*d/dx)^n when n is even.

%I A223168 #73 Dec 25 2023 18:01:25
%S A223168 1,1,2,3,2,3,12,4,15,20,4,15,90,60,8,105,210,84,8,105,840,840,224,16,
%T A223168 945,2520,1512,288,16,945,9450,12600,5040,720,32,10395,34650,27720,
%U A223168 7920,880,32,10395,124740,207900,110880,23760,2112,64,135135,540540,540540,205920,34320,2496,64
%N A223168 Triangle S(n, k) by rows: coefficients of 2^((n-1)/2)*(x^(1/2)*d/dx)^n when n is odd, and of 2^(n/2)*(x^(1/2)*d/dx)^n when n is even.
%C A223168 Also coefficients in the expansion of k-th derivative of exp(n*x^2), see Mathematica program. - _Vaclav Kotesovec_, Jul 16 2013
%H A223168 Vincenzo Librandi, <a href="/A223168/b223168.txt">Rows n = 0..60, flattened</a>
%H A223168 U. N. Katugampola, <a href="http://dx.doi.org/10.1016/j.amc.2014.12.067">Mellin Transforms of Generalized Fractional Integrals and Derivatives</a>, Appl. Math. Comput. 257(2015) 566-580.
%H A223168 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 [math.CA], 2014.
%e A223168 Triangle begins:
%e A223168        1;
%e A223168        1,      2;
%e A223168        3,      2;
%e A223168        3,     12,      4;
%e A223168       15,     20,      4;
%e A223168       15,     90,     60,      8;
%e A223168      105,    210,     84,      8;
%e A223168      105,    840,    840,    224,    16;
%e A223168      945,   2520,   1512,    288,    16;
%e A223168      945,   9450,  12600,   5040,   720,   32;
%e A223168    10395,  34650,  27720,   7920,   880,   32;
%e A223168    10395, 124740, 207900, 110880, 23760, 2112, 64;
%e A223168   135135, 540540, 540540, 205920, 34320, 2496, 64;
%e A223168   .
%e A223168 Expansion takes the form:
%e A223168 2^0 (x^(1/2)*d/dx)^1 = 1*x^(1/2)*d/dx.
%e A223168 2^1 (x^(1/2)*d/dx)^2 = 1*d/dx + 2*x*d^2/dx^2.
%e A223168 2^1 (x^(1/2)*d/dx)^3 = 3*x^(1/2)*d^2/dx^2 + 2*x^(3/2)*d^3/dx^3.
%e A223168 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.
%e A223168 2^2 (x^(1/2)*d/dx)^5 = 15*x^(1/2)*d^3/dx^3 + 20*x^(3/2)*d^4/dx^4 + 4*x^(5/2)*d^5/dx^5.
%e A223168 `
%e A223168 `
%p A223168 a[0]:= f(x);
%p A223168 for i from 1 to 13 do
%p A223168 a[i]:= simplify(2^((i+1)mod 2)*x^(1/2)*(diff(a[i-1],x$1)));
%p A223168 end do;
%t A223168 Flatten[CoefficientList[Expand[FullSimplify[Table[D[E^(n*x^2),{x,k}]/(E^(n*x^2)*(2*n)^Floor[(k+1)/2]),{k,1,13}]]]/.x->1,n]] (* _Vaclav Kotesovec_, Jul 16 2013 *)
%Y A223168 Odd rows includes absolute values of A098503 from right to left.
%Y A223168 Cf. A223169-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.
%K A223168 nonn,tabf
%O A223168 0,3
%A A223168 _Udita Katugampola_, Mar 17 2013