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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A223169 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)*(x^(1/3)*d/dx)^n when n is odd, and of 3^(n/2)*(x^(2/3)*d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 3, 4, 3, 4, 24, 9, 28, 42, 9, 28, 252, 189, 27, 280, 630, 270, 27, 280, 3360, 3780, 1080, 81, 3640, 10920, 7020, 1404, 81, 3640, 54600, 81900, 35100, 5265, 243, 58240, 218400, 187200, 56160, 6480, 243, 58240, 1048320, 1965600
Offset: 0

Views

Author

Udita Katugampola, Mar 18 2013

Keywords

Examples

			Triangle begins:
1;
1, 3;
4, 3;
4, 24, 9;
28, 42, 9;
28, 252, 189, 27;
280, 630, 270, 27;
280, 3360, 3780, 1080, 81;
3640, 10920, 7020, 1404, 81;
3640, 54600, 81900, 35100, 5265, 243,
58240, 218400, 187200, 56160, 6480, 243

Links

Crossrefs

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

Programs

  • Maple
    a[0]:= f(x):
for i from 1 to 13 do
a[i] := simplify(3^((i+1)mod 2)*x^(((i+1)mod 2+1)/3)*(diff(a[i-1],x$1 )));
end do;

A223170 Triangle S(n,k) by rows: coefficients of 4^((n-1)/2)*(x^(1/4)*d/dx)^n when n is odd, and of 4^(n/2)*(x^(3/4)*d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 4, 5, 4, 5, 40, 16, 45, 72, 16, 45, 540, 432, 64, 585, 1404, 624, 64, 585, 9360, 11232, 3328, 256, 9945, 31824, 21216, 4352, 256, 9945, 198900, 318240, 141440, 21760, 1024, 208845, 835380, 742560, 228480, 26880, 1024, 208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096
Offset: 0

Views

Author

Udita Katugampola, Mar 20 2013

Keywords

Examples

			Triangle begins:
1;
1, 4;
5, 4;
5, 40, 16;
45, 72, 16;
45, 540, 432, 64;
585, 1404, 624, 64;
585, 9360, 11232, 3328, 256;
9945, 31824, 21216, 4352, 256;
9945, 198900, 318240, 141440, 21760, 1024;
208845, 835380, 742560, 228480, 26880, 1024;
208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096;

Links

Crossrefs

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

Programs

  • Maple
    a[0]:= f(x):
for i from 1 to 13 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;
  • Mathematica
    nmax = 12;
b[0] = Exp[x]; For[ i = 1 , i <= nmax , i++, b[i] = 4^Mod[i + 1, 2]*x^((2 Mod[i + 1, 2] + 1)/4)*D[b[i - 1], x]] // Simplify;
row[1] = {1}; row[n_] := List @@ Expand[b[n]/f[x]] /. x -> 1;
Table[row[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Feb 22 2019, from Maple *)

Extensions

Missing terms inserted by Jean-François Alcover, Feb 22 2019

A223171 Triangle S(n,k) by rows: coefficients of 5^((n-1)/2)*(x^(1/5)*d/dx)^n when n is odd, and of 5^(n/2)*(x^(4/5)*d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 5, 6, 5, 6, 60, 25, 66, 110, 25, 66, 990, 825, 125, 1056, 2640, 1200, 125, 1056, 21120, 26400, 8000, 625, 22176, 73920, 50400, 10500, 625, 22176, 554400, 924000, 420000, 65625, 3125, 576576, 2402400, 2184000, 682500, 81250, 3125, 576576, 17297280
Offset: 0

Views

Author

Udita Katugampola, Mar 20 2013

Keywords

Examples

			Triangle begins:
1;
1, 5;
6, 5;
6, 60, 25;
66, 110, 25;
66, 990, 825, 125;
1056, 2640, 1200, 125;
1056, 21120, 26400, 8000, 625;
22176, 73920, 50400, 10500, 625;
22176, 554400, 924000, 420000, 65625, 3125;
576576, 2402400, 2184000, 682500, 81250, 3125;
576576, 17297280, 36036000, 21840000, 5118750, 487500, 15625;
17873856, 89369280, 101556000, 42315000, 7556250, 581250, 15625;

Links

Crossrefs

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

Programs

  • Maple
    a[0]:= f(x):
for i from 1 to 13 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;

A132060 Row sums of triangle A132056 (S2(8), Stirling2 generalization).

Original entry on oeis.org

1, 9, 145, 3361, 101601, 3786601, 167756689, 8611201665, 502522701121, 32854723149961, 2378687990172561, 188913154677765409, 16328988725252964385, 1525987835722540609641, 153312690475268723848081
Offset: 1

Views

Author

Wolfdieter Lang Sep 14 2007

Keywords

Comments

Generalized Bell numbers B(8,1;n).

Crossrefs

Cf. A092084 (row sums of S2(7)=A092082).

Programs

  • Mathematica
    With[{nn=30},Rest[CoefficientList[Series[Exp[-1+(1-7x)^(-1/7)]-1,{x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Sep 23 2011 *)

Formula

a(n)=sum(A132056(n, m), m=1..n), n>=1.
E.g.f.: exp(-1+(1-7*x)^(-1/7)) - 1.

A132061 Alternating row sums of triangle A132056 (S2(8), Stirling2 generalization).

Original entry on oeis.org

1, 7, 97, 2015, 55841, 1935719, 80574433, 3915183103, 217530794305, 13603055116679, 945542295992801, 72321915976403807, 6036466379411066977, 545983089637963491175, 53194885608199879974241
Offset: 0

Views

Author

Wolfdieter Lang Sep 14 2007

Keywords

Crossrefs

Cf. A092085 (alternating row sums of S2(7)=A092082).

Formula

a(n)= -sum(A132056(n, m)*(-1)^m, m=1..n), n>=1.
E.g.f.:-(exp(1 - (1-7*x)^(-1/7)) - 1).

A223512 Triangle T(n,k) represents the coefficients of (x^10*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 10, 1, 190, 30, 1, 5320, 1060, 60, 1, 196840, 45600, 3400, 100, 1, 9054640, 2340040, 208800, 8300, 150, 1, 498005200, 140096880, 14241640, 690200, 17150, 210, 1, 31872332800, 9604302400, 1080045120, 60485040, 1856400, 31640, 280, 1, 2326680294400
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			1;
10,1;
190,30,1;
5320,1060,60,1;
196840,45600,3400,100,1;
9054640,2340040,208800,8300,150,1;
498005200,140096880,14241640,690200,17150,210,1;
31872332800,9604302400,1080045120,60485040,1856400,31640,280,1,2326680294400

Crossrefs

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

Programs

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

A223513 Triangle T(n,k) represents the coefficients of (x^11*d/dx)^n, where n=1,2,3,...

Original entry on oeis.org

1, 11, 1, 231, 33, 1, 7161, 1287, 66, 1, 293601, 61215, 4125, 110, 1, 14973651, 3476781, 279840, 10065, 165, 1, 913392711, 230534073, 21106701, 924000, 20790, 231, 1, 64850882481, 17511845967, 1771323246, 89482701, 2483250, 38346, 308, 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;
11,1;
231,33,1;
7161,1287,66,1;
293601,61215,4125,110,1;
14973651,3476781,279840,10065,165,1;
913392711,230534073,21106701,924000,20790,23,1;
64850882481,17511845967,1771323246,89482701,2483250,38346,308,1;

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^11*diff(b[j-1],x$1);
end do;

A223514 Triangle T(n,k) represents the coefficients of (x^12*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 12, 1, 276, 36, 1, 9384, 1536, 72, 1, 422280, 80040, 4920, 120, 1, 23647680, 4984560, 365400, 12000, 180, 1, 1584394560, 362597760, 30197160, 1205400, 24780, 252, 1, 123582775680, 30229617600, 2778370560, 127834560, 3237360, 45696, 336, 1, 1099867035520
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			1;
12,1;
276,36,1;
9384,1536,72,1;
422280,80040,4920,120,1;
23647680,4984560,365400,12000,180,1;
1584394560,362597760,30197160,1205400,24780,252,1;
123582775680,30229617600,2778370560,127834560,3237360,45696,336,1;
1099867035520,...

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^12*diff(b[j-1],x$1);
end do;

A223515 Triangle T(n,k) represents the coefficients of (x^13*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 13, 1, 325, 39, 1, 12025, 1807, 78, 1, 589225, 102375, 5785, 130, 1, 35942725, 6936475, 466830, 14105, 195, 1, 2623818925, 549241875, 41948725, 1538810, 29120, 273, 1, 223024608625, 49858620175, 4198780950, 177364005, 4130490, 53690, 364, 1, 21633387036625
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			1;
13,1;
325,39,1;
12025,1807,78,1;
589225,102375,5785,130,1;
35942725,6936475,466830,14105,195,1
2623818925,549241875,41948725,1538810,29120,273,1;
223024608625,49858620175,4198780950,177364005,4130490,53690,364,1;
21633387036625,...

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^13*diff(b[j-1],x$1);
end do;

A223516 Triangle T(n,k) represents the coefficients of (x^14*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 14, 1, 378, 42, 1, 15120, 2100, 84, 1, 801360, 128520, 6720, 140, 1, 52889760, 9412200, 585480, 16380, 210, 1, 4178291040, 805865760, 56836080, 1928640, 33810, 294, 1, 384402775680, 79123806720, 6148457280, 240056880, 5174400, 62328, 392, 1
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			1;
14,1;
378,42,1;
15120,2100,84,1;
801360,128520,6720,140,1;
52889760,9412200,585480,16380,210,1;
4178291040,805865760,56836080,1928640,33810,294,1;
384402775680,79123806720,6148457280,240056880,5174400,62328,392,1;

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^14*diff(b[j-1],x$1);
end do;
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