A223169 -id:A223169 - cp's OEIS frontend

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.

1, 1, 2, 3, 2, 3, 12, 4, 15, 20, 4, 15, 90, 60, 8, 105, 210, 84, 8, 105, 840, 840, 224, 16, 945, 2520, 1512, 288, 16, 945, 9450, 12600, 5040, 720, 32, 10395, 34650, 27720, 7920, 880, 32, 10395, 124740, 207900, 110880, 23760, 2112, 64, 135135, 540540, 540540, 205920, 34320, 2496, 64

Offset: 0

Udita Katugampola, Mar 17 2013

Also coefficients in the expansion of k-th derivative of exp(n*x^2), see Mathematica program. - Vaclav Kotesovec, Jul 16 2013

			Triangle begins:
       1;
       1,      2;
       3,      2;
       3,     12,      4;
      15,     20,      4;
      15,     90,     60,      8;
     105,    210,     84,      8;
     105,    840,    840,    224,    16;
     945,   2520,   1512,    288,    16;
     945,   9450,  12600,   5040,   720,   32;
   10395,  34650,  27720,   7920,   880,   32;
   10395, 124740, 207900, 110880, 23760, 2112, 64;
  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)^2 = 1*d/dx + 2*x*d^2/dx^2.
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)^4 = 3*d^2/dx^2 + 12*x*d^3/dx^3 + 4*x^2*d^4/dx^4.
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.
`
`

Vincenzo Librandi, Rows n = 0..60, flattened
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 [math.CA], 2014.

Odd rows includes absolute values of A098503 from right to left.

Cf. A223169-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

a[0]:= f(x);
for i from 1 to 13 do
a[i]:= simplify(2^((i+1)mod 2)*x^(1/2)*(diff(a[i-1],x$1)));
end do;

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 *)

A223525 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)(x^(1/3)d/dx)^n when n=1,3,5,...

Original entry on oeis.org

1, 4, 3, 4, 24, 9, 28, 252, 189, 27, 280, 3360, 3780, 1080, 81, 3640, 54600, 81900, 35100, 5265, 243, 1106560, 4979520, 5335200, 2134080, 369360, 27702, 729, 24344320, 127807680, 164324160, 82162080, 18960480, 2133054, 112266, 2187, 608608000

Offset: 1

Udita Katugampola, Mar 18 2013

			Triangle begins:
1;
4, 3;
4, 24, 9;,
28, 252, 189, 27;
280, 3360, 3780, 1080, 81;
3640, 54600, 81900, 35100, 5265, 243;
1106560, 4979520, 5335200, 2134080, 369360, 27702, 729;
24344320, 127807680, 164324160, 82162080, 18960480, 2133054, 112266, 2187;

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

Odd rows of A223169.

Cf. A223168-A223172, A223511-A223532.

a[0]:= f(x):
for i from 1 to 20 do
a[i] := simplify(3^((i+1)mod 2)*x^(((i+1)mod 2+1)/3)*(diff(a[i-1],x$1 )));
end do:
for j from 1 to 10 do
b[j]:=a[2j-1];
end do;

A223526 Triangle S(n,k) by rows: coefficients of 3^(n/2)(x^(2/3)d/dx)^n when n=0,2,4,6,...

Original entry on oeis.org

1, 1, 3, 4, 24, 9, 28, 252, 189, 27, 280, 3360, 3780, 1080, 81, 3640, 54600, 81900, 35100, 5265, 243, 58240, 1048320, 1965600, 1123200, 252720, 23328, 729, 1106560, 23237760, 52284960, 37346400, 11203920, 1551312, 96957, 2187, 24344320, 584263680, 1533692160

Offset: 1

Udita Katugampola, Mar 18 2013

			Triangle begins:
1;
1, 3;
4, 24, 9;
28, 252, 189, 27;
280, 3360, 3780, 1080, 81;
3640, 54600, 81900, 35100, 5265, 243;
58240, 1048320, 1965600, 1123200, 252720, 23328, 729;
1106560, 23237760, 52284960, 37346400, 11203920, 1551312, 96957, 2187;
24344320, 584263680, 1533692160, 1314593280, 492972480, 91010304, 8532216, 384912, 6561;

U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.

Even row of A223169.

Cf. A223168-A223172, A223511-A223532.

a[0]:= f(x):
for i from 1 to 20 do
a[i] := simplify(3^((i+1)mod 2)*x^(((i+1)mod 2+1)/3)*(diff(a[i-1],x$1 )));
end do:
for j from 1 to 10 do
b[j]:=a[2j];
end do;

T(n,0) = A007559(n) and T(n,n) = A000244(n) for all n>=0

cp's OEIS Frontend

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.

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A223525 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)(x^(1/3)d/dx)^n when n=1,3,5,...

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A223526 Triangle S(n,k) by rows: coefficients of 3^(n/2)(x^(2/3)d/dx)^n when n=0,2,4,6,...

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cp's OEIS Frontend

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.

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Maple

Mathematica

A223525 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)*(x^(1/3)*d/dx)^n when n=1,3,5,...

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A223526 Triangle S(n,k) by rows: coefficients of 3^(n/2)*(x^(2/3)*d/dx)^n when n=0,2,4,6,...

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

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.

A223525 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)(x^(1/3)d/dx)^n when n=1,3,5,...

A223526 Triangle S(n,k) by rows: coefficients of 3^(n/2)(x^(2/3)d/dx)^n when n=0,2,4,6,...