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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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;

A371080 Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 0, 28, 12, 1, 0, 280, 160, 24, 1, 0, 3640, 2520, 520, 40, 1, 0, 58240, 46480, 11880, 1280, 60, 1, 0, 1106560, 987840, 295960, 40040, 2660, 84, 1, 0, 24344320, 23826880, 8090880, 1296960, 109200, 4928, 112, 1
Offset: 0

Views

Author

Peter Luschny, Mar 12 2024

Keywords

Examples

			Triangle starts:
[0] 1;
[1] 0,       1;
[2] 0,       4,      1;
[3] 0,      28,     12,      1;
[4] 0,     280,    160,     24,     1;
[5] 0,    3640,   2520,    520,    40,    1;
[6] 0,   58240,  46480,  11880,  1280,   60,  1;
[7] 0, 1106560, 987840, 295960, 40040, 2660, 84, 1;

Links

Crossrefs

Variant: A035469.
Cf. A264428, A371076, A371077.
Cf. A007559, A144346.
Cf. A035342, A049029, A271703.

Programs

  • Maple
    a := n -> mul(select(k -> k mod 3 = 1, [seq(1..3*(n + 1))])): BellMatrix(a, 9);
# Alternative:
BellMatrix(n -> coeff(series((1/x)*hypergeom([1, 1/3], [], 3*x),x, 22), x, n), 9);
# Recurrence:
T := proc(n, k) option remember; if k = n then 1 elif k = 0 then 0 else
T(n - 1, k - 1) + (3*(n - 1) + k) * T(n - 1, k) fi end:
for n from 0 to 7 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Mar 13 2024
  • PARI
    T(n, k) = sum(j=k, n, 3^(n-j)*abs(stirling(n, j, 1))*stirling(j, k, 2)); \\ Seiichi Manyama, Apr 19 2025

Formula

T(n, k) = BellMatrix([x^n] hypergeom2F0([1, 1/3], [], 3*x) / x).
T(n, k) = A371076(n, k) / k!.
From Werner Schulte, Mar 13 2024: (Start)
T(n, k) = (Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * Product_{j=0..n-1} (3*j + i)) / (k!).
T(n, k) = T(n-1, k-1) + (3*(n - 1) + k) * T(n-1, k) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = 1 for n >= 0. (End)
From Seiichi Manyama, Apr 19 2025: (Start)
T(n,k) = Sum_{j=k..n} 3^(n-j) * |Stirling1(n,j)| * Stirling2(j,k).
E.g.f. of column k (with leading zeros): (1/(1 - 3*x)^(1/3) - 1)^k / k!. (End)

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;

A223517 Triangle T(n,k) represents the coefficients of (x^15*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, 15, 1, 435, 45, 1, 18705, 2415, 90, 1, 1066185, 158775, 7725, 150, 1, 75699135, 12497985, 722700, 18825, 225, 1, 6434426475, 1150525845, 75372885, 2379300, 38850, 315, 1, 637008221025, 121487010975, 8763187230, 318061485, 6380850, 71610, 420, 1
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			1;
15,1;
435,45,1;
18705,2415,90,1;
1066185,158775,7725,150,1;
75699135,12497985,722700,18825,225,1;
6434426475,1150525845,75372885,2379300,38850,315,1;
637008221025,121487010975,8763187230,318061485,6380850,71610,420,1;

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^15*diff(b[j-1],x$1);
end do;

A223518 Triangle T(n,k) represents the coefficients of (x^16*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, 16, 1, 496, 48, 1, 22816, 2752, 96, 1, 1391776, 193440, 8800, 160, 1, 105774976, 16286656, 879840, 21440, 240, 1, 9625522816, 1604147328, 98111776, 2895200, 44240, 336, 1, 1020305418496, 181269286912, 12200219136, 413688576, 7761600, 81536, 448, 1
Offset: 1

Views

Author

Udita Katugampola, Mar 23 2013

Keywords

Examples

			1;
16,1;
496,48,1;
22816,2752,96,1;
1391776,193440,8800,160,1;
105774976,16286656,879840,21440,240,1;
9625522816,1604147328,98111776,2895200,44240,336,1;
1020305418496,181269286912,12200219136,413688576,7761600,81536,448,1

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^16*diff(b[j-1],x$1);
end do;

A223519 Triangle T(n,k) represents the coefficients of (x^17*d/dx)^n, where n=1,2,3,...

Original entry on oeis.org

1, 17, 1, 561, 51, 1, 27489, 3111, 102, 1, 1786785, 232815, 9945, 170, 1, 144729585, 20877615, 1058250, 24225, 255, 1, 14038769745, 2190735855, 125644365, 3480750, 49980, 357, 1, 1586380981185, 263782657215, 16639837830, 529411365, 9328410, 92106, 476, 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;
17,1;
561,51,1;
27489,3111,102,1;
1786785,232815,9945,170,1;
144729585,20877615,1058250,24225,255,1;
14038769745,2190735855,125644365,3480750,49980,357,1;
1586380981185,263782657215,16639837830,529411365,9328410,92106,476,1;

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^17*diff(b[j-1],x$1);
end do;
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