A223536 -id:A223536 - cp's OEIS frontend

A223533 Coefficients of (x^(1/3)*d/dx)^n for positive integer n.

1, 1, 3, -1, 9, 9, 1, -1, 18, 9, -5, 5, 15, 90, 27, 35, -35, 225, 405, 81, -105, 105, -35, 630, 567, 81, 1155, -1155, 490, -105, 4158, 2268, 243, 15015, -15015, 6895, 945, -10206, -23814, -8748, -729, 75075, -75075, 35700, -10675, 2835, -945, 34020, 41310, 10935, 729

Offset: 1

Udita Katugampola, Apr 18 2013

These are generalized Stirling numbers.

			1;
1, 3;
-1, 9, 9;
1, -1, 18, 9;
-5, 5, 15, 90, 27;
35, -35, 225, 405, 81;
-105, 105, 630, 567, -35, 81;
1155, -1155, 630, 4158, 490, 2268, -105, 243;

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

Cf. A223168-A223172, A223534-A223536.

# This will generate the sequence as coefficients of pseudo polynomials
# up to a constant multiple.
a[0] := f(x):
for i to 10 do
a[i] := simplify(x^(1/3)*(diff(a[i-1],x$1)))
end do;

G.f.: exp(((1+2/3*x*y)^(3/2)-1)/x).

A223534 Coefficients of (x^(1/4)*d/dx)^n for n positive integer.

Original entry on oeis.org

1, 1, 4, -1, 6, 8, 5, -10, 48, 32, -10, 15, -10, 80, 32, 110, -145, 90, 40, 480, 128, -770, 945, -560, 140, 560, 1344, 256, -13090, 15365, -8820, 2940, -6272, -7168, -1024, 65450, -74550, 41825, -14700, 2940, -13440, -9216, -1024, 1505350, -1678250, 925575

Offset: 1

Udita Katugampola, Apr 18 2013

These are generalized Stirling numbers.

			1;
1, 4;
-1, 6, 8;
5, -10, 48, 32;
-10, 15, -10, 80, 32;
110, -145, 90, 40, 480, 128;
-770, 945, -560, 140, 560, 1344, 256;

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

Cf. A223168-A223172, A223533-A223536.

# This will generate the sequence as coefficients of pseudo polynomials
# up to a constant multiple.
a[0] := f(x):
for i to 10 do
a[i] := simplify(x^(1/4)*(diff(a[i-1],x$1)))
end do;

G.f.: exp(((1+3/4*x*y)^(4/3)-1)/x).

A223535 Coefficients of (x^(1/5)*d/dx)^n for positive integer n.

Original entry on oeis.org

1, 1, 5, -3, 15, 25, 21, -45, 150, 125, -231, 375, -375, 1250, 625, 693, -981, 750, -375, 1875, 625, -13167, 17199, -11655, 5250, 13125, 3125, 302841, -375417, 237510, -100275, 26250, 26250, 87500, 15625, 8176707, -9773379, 5914755, -2390850, 685125, -78750

Offset: 1

Udita Katugampola, Apr 18 2013

These are generalized Stirling numbers.

			1;
1, 5;
-3, 15, 25;
21, -45, 150, 125;
-231, 375, -375, 1250, 625;
693, -981, 750, -375, 1875, 625;

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

Cf. A223168-A223172, A223533-A223536.

# This will generate the sequence as coefficients of pseudo polynomials
# up to a constant multiple.
a[0] := f(x):
for i to 10 do
a[i] := simplify(x^(1/5)*(diff(a[i-1],x$1)))
end do;

G.f.: exp(((1+4/5*x*y)^(5/4)-1)/x).

cp's OEIS Frontend

A223533 Coefficients of (x^(1/3)*d/dx)^n for positive integer n.

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A223534 Coefficients of (x^(1/4)*d/dx)^n for n positive integer.

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A223535 Coefficients of (x^(1/5)*d/dx)^n for positive integer n.

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Keywords

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Examples

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Crossrefs

Programs

Maple

Formula