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 A223523 Triangle S(n, k) by rows: coefficients of 2^((n-1)/2))*(x^(1/2)*d/dx)^n, where n = 1, 3, 5, ... 25
 1, 3, 2, 15, 20, 4, 105, 210, 84, 8, 945, 2520, 1512, 288, 16, 10395, 34650, 27720, 7920, 880, 32, 135135, 540540, 540540, 205920, 34320, 2496, 64, 2027025, 9459450, 11351340, 5405400, 1201200, 131040, 6720, 128 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Triangle S(n,n-k) by rows: coefficients of 2^n * |L(n,1/2,x)|, with L the generalized Laguerre polynomials. LINKS 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 FORMULA T(n, k) = 2^n * n!/(n-k)! * C(n+1/2, k), n>=0, k<=n. EXAMPLE Triangle begins: 1; 3, 2; 15, 20, 4; 105, 210, 84, 8; 945, 2520, 1512, 288, 16; 10395, 34650, 27720, 7920, 880, 32; 135135, 540540, 540540, 205920, 34320, 2496, 64; . . Expansion takes the form: 2^0 (x^(1/2)*d/dx)^1 = 1*x^(1/2)*d/dx. 2^1 (x^(1/2)*d/dx)^3 = 3*x^(1/2)*d^2/dx^2 + 2*x^(3/2)*d^3/dx^3. 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. 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-1]; end do; CROSSREFS Cf. A223168-A223172. Rows includes odd rows of A223168. Rows includes absolute values of A098503 from right to left of the triangular form. Sequence in context: A051917 A302845 A291251 * A133932 A111999 A286947 Adjacent sequences:  A223520 A223521 A223522 * A223524 A223525 A223526 KEYWORD nonn,tabl AUTHOR Udita Katugampola, Mar 21 2013 STATUS approved

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Last modified August 2 08:13 EDT 2021. Contains 346422 sequences. (Running on oeis4.)