cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A223520 Triangle T(n,k) represents the coefficients of (x^18*d/dx)^n, where n=1,2,3,....

Original entry on oeis.org

1, 18, 1, 630, 54, 1, 32760, 3492, 108, 1, 2260440, 277200, 11160, 180, 1, 194397840, 26376840, 1259280, 27180, 270, 1, 20022977520, 2937589200, 158601240, 4140360, 56070, 378, 1, 2402757302400, 375471270720, 22286940480, 667865520, 11093040, 103320, 504, 1
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Comments

Generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Examples

			1;
18,1;
630,54,1;
32760,3492,108,1;
2260440,277200,11160,180,1;
194397840,26376840,1259280,27180,270,1;
20022977520,2937589200,158601240,4140360,56070,378,1;
2402757302400,375471270720,22286940480,667865520,11093040,103320,504,1

Crossrefs

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^18*diff(b[j-1],x$1);
end do;

A223521 Triangle T(n,k) represents the coefficients of (x^19*d/dx)^n, where n=1,2,3,...

Original entry on oeis.org

1, 19, 1, 703, 57, 1, 38665, 3895, 114, 1, 2822545, 326895, 12445, 190, 1, 256851595, 32896885, 1484280, 30305, 285, 1, 27996823855, 3875508945, 197651965, 4878440, 62510, 399, 1, 3555596629585, 524061968815, 29372612430, 831849165, 13067250, 115178, 532, 1
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Comments

Generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Examples

			1;
19,1;
703,57,1;
38665,3895,114,1;
2822545,326895,12445,190,1;
256851595,32896885,1484280,30305,285,1;
27996823855,3875508945,197651965,4878440,62510,399,1;
3555596629585,524061968815,29372612430,831849165,13067250,115178,532,1;

Crossrefs

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^19*diff(b[j-1],x$1);
end do;

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

Original entry on oeis.org

1, 1, 2, 3, 12, 4, 15, 90, 60, 8, 105, 840, 840, 224, 16, 945, 9450, 12600, 5040, 720, 32, 10395, 124740, 207900, 110880, 23760, 2112, 64, 135135, 1891890, 3783780, 2522520, 720720, 96096, 5824, 128, 2027025, 32432400
Offset: 1

Views

Author

Udita Katugampola, Mar 21 2013

Keywords

Comments

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

Examples

			Triangle begins:
  1;
  1, 2;
  3, 12, 4;
  15, 90, 60, 8;
  105, 840, 840, 224, 16;
  945, 9450, 12600, 5040, 720, 32;
  10395, 124740, 207900, 110880, 23760, 2112, 64;
  ...
Expansion takes the form:
  2^1 (x^(1/2)*d/dx)^2 = 1*d/dx + 2*x*d^2/dx^2.
  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.

Links

Crossrefs

Rows includes even rows of A223168.
Cf. A223168-A223172 and A223523.

Programs

  • 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];
end do;
  • Mathematica
    Flatten[Abs[Table[CoefficientList[2^n n! LaguerreL[n, -1/2, x], x], {n, 0, 7}]]] (* Ali Pourzand, Mar 28 2025 *)

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

Views

Author

Udita Katugampola, Mar 18 2013

Keywords

Examples

			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;

Links

Crossrefs

Odd rows of A223169.
Cf. A223168-A223172, A223511-A223532.

Programs

  • Maple
    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

Views

Author

Udita Katugampola, Mar 18 2013

Keywords

Examples

			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;

Links

Crossrefs

Even row of A223169.
Cf. A223168-A223172, A223511-A223532.

Programs

  • Maple
    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;

Formula

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

A223527 Triangle S(n,k) by rows: coefficients of 4^((n-1)/2)*(x^(1/4)*d/dx)^n when n=1,3,5,...

Original entry on oeis.org

1, 5, 4, 45, 72, 16, 585, 1404, 624, 64, 9945, 31824, 21216, 4352, 256, 208845, 835380, 742560, 228480, 26880, 1024, 5221125, 25061400, 27846000, 11424000, 2016000, 153600, 4096, 151412625, 847910700, 1130547600, 579768000, 136416000, 15590400, 831488, 16384
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			Triangle begins:
1;
5, 4;
45, 72, 16;
585, 1404, 624, 64;
9945, 31824, 21216, 4352, 256;
208845, 835380, 742560, 228480, 26880, 1024;
5221125, 25061400, 27846000, 11424000, 2016000, 153600, 4096;
151412625, 847910700, 1130547600, 579768000, 136416000, 15590400, 831488, 16384;

Links

Crossrefs

Odd rows of A223170.
Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    a[0]:= f(x):
for i from 1 to 20 do
a[i] := simplify(4^((i+1)mod 2)*x^((2((i+1)mod 2)+1)/4)*(diff(a[i-1],x$1 )));
end do:
for j from 1 to 10 do
b[j]:=a[2j-1];
end do;

A223528 Triangle S(n,k) by rows: coefficients of 4^(n/2)*(x^(3/4)*d/dx)^n when n=0,2,4,6,...

Original entry on oeis.org

1, 1, 4, 5, 40, 16, 45, 540, 432, 64, 585, 9360, 11232, 3328, 256, 9945, 198900, 318240, 141440, 21760, 1024, 208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096, 5221125, 146191500, 350859600, 259896000, 79968000, 11289600, 716800, 16384, 151412625
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			Triangle begins:
1;
1, 4;
5, 40, 16;
45, 540, 432, 64;
585, 9360, 11232, 3328, 256;
9945, 198900, 318240, 141440, 21760, 1024;
208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096;
5221125, 146191500, 350859600, 259896000, 79968000, 11289600, 716800, 16384;
151412625, 4845204000, 13566571200, 12059174400, 4638144000, 873062400, 83148800, 3801088, 65536;

Links

Crossrefs

Even rows of A223170.
Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    a[0]:= f(x):
for i from 1 to 20 do
a[i] := simplify(4^((i+1)mod 2)*x^((2((i+1)mod 2)+1)/4)*(diff(a[i-1],x$1 )));
end do:
for j from 1 to 10 do
b[j]:=a[2j];
end do;

A223529 Triangle S(n,k) by rows: coefficients of 5^((n-1)/2)*(x^(1/5)*d/dx)^n when n=1,3,5,...

Original entry on oeis.org

1, 6, 5, 66, 110, 25, 1056, 2640, 1200, 125, 22176, 73920, 50400, 10500, 625, 576576, 2402400, 2184000, 682500, 81250, 3125, 17873856, 89369280, 101556000, 42315000, 7556250, 581250, 15625, 643458816, 3753509760, 5118422400, 2665845000, 634725000
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			Triangle begins:
1;
6, 5;
66, 110, 25;
1056, 2640, 1200, 125;
22176, 73920, 50400, 10500, 625;
576576, 2402400, 2184000, 682500, 81250, 3125;
17873856, 89369280, 101556000, 42315000, 7556250, 581250, 15625;
643458816, 3753509760, 5118422400, 2665845000, 634725000, 73237500, 3937500, 78125;

Links

Crossrefs

Odd rows of A223171.
Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    a[0]:= f(x):
for i from 1 to 20 do
a[i] := simplify(5^((i+1)mod 2)*x^((3((i+1)mod 2)+1)/5)*(diff(a[i-1],x$1 )));
end do:
for j from 1 to 10 do
b[j]:=a[2j-1];
end do;

A223530 Triangle S(n,k) by rows: coefficients of 5^(n/2)*(x^(4/5)*d/dx)^n when n=0,2,4,6,...

Original entry on oeis.org

1, 1, 5, 6, 60, 25, 66, 990, 825, 125, 1056, 21120, 26400, 8000, 625, 22176, 554400, 924000, 420000, 65625, 3125, 576576, 17297280, 36036000, 21840000, 5118750, 487500, 15625, 17873856, 625584960, 1563962400, 1184820000, 370256250, 52893750, 3390625, 78125
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			Triangle begins:
1;
1, 5;
6, 60, 25;
66, 990, 825, 125;
1056, 21120, 26400, 8000, 625;
22176, 554400, 924000, 420000, 65625, 3125;
576576, 17297280, 36036000, 21840000, 5118750, 487500, 15625;
17873856, 625584960, 1563962400, 1184820000, 370256250, 52893750, 3390625, 78125;

Links

Crossrefs

Even rows of A223171.
Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    a[0]:= f(x):
for i from 1 to 20 do
a[i] := simplify(5^((i+1)mod 2)*x^((3((i+1)mod 2)+1)/5)*(diff(a[i-1],x$1 )));
end do:
for j from 1 to 10 do
b[j]:=a[2j];
end do;

A223531 Triangle S(n,k) by rows: coefficients of 6^((n-1)/2)*(x^(1/6)*d/dx)^n when n=1,3,5,...

Original entry on oeis.org

1, 7, 6, 91, 156, 36, 1729, 4446, 2052, 216, 43225, 148200, 102600, 21600, 1296, 1339975, 5742750, 5301000, 1674000, 200880, 7776, 49579075, 254978100, 294205500, 123876000, 22297680, 1726272, 46656, 2131900225, 12791401350, 17711171100, 9321669000
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			Triangle begins:
1;
7, 6;
91, 156, 36;
1729, 4446, 2052, 216;
43225, 148200, 102600, 21600, 1296;
1339975, 5742750, 5301000, 1674000, 200880, 7776;
49579075, 254978100, 294205500, 123876000, 22297680, 1726272, 46656;
2131900225, 12791401350, 17711171100, 9321669000, 2237200560, 259803936, 14043456, 279936;

Links

Crossrefs

Odd rows of A223172.
Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

Programs

  • Maple
    a[0]:= f(x):
for i from 1 to 20 do
a[i] := simplify(6^((i+1)mod 2)*x^((4((i+1)mod 2)+1)/6)*(diff(a[i-1],x$1 )));
end do:
for j from 1 to 10 do
b[j]:=a[2j-1];
end do;
Previous Showing 21-30 of 31 results. Next

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