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.

Previous Showing 11-19 of 19 results.

A176627 Triangle T(n, k) = 12^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 144, 144, 1, 1, 1728, 20736, 1728, 1, 1, 20736, 2985984, 2985984, 20736, 1, 1, 248832, 429981696, 5159780352, 429981696, 248832, 1, 1, 2985984, 61917364224, 8916100448256, 8916100448256, 61917364224, 2985984, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 22 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      12,           1;
  1,     144,         144,             1;
  1,    1728,       20736,          1728,             1;
  1,   20736,     2985984,       2985984,         20736,           1;
  1,  248832,   429981696,    5159780352,     429981696,      248832,       1;
  1, 2985984, 61917364224, 8916100448256, 8916100448256, 61917364224, 2985984, 1;

Links

Crossrefs

Cf. A000326,
Cf. A118190 (q=2), this sequence (q=3), A176631 (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), this sequence (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).

Programs

  • Magma
    [(12)^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 30 2021
  • Mathematica
    (* First program *)
T[n_, k_, q_]= (Binomial[3*q,2]/3)^(k*(n-k));
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 30 2021 *)
(* Second program *)
With[{m=10}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)
  • Sage
    flatten([[(12)^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

Formula

T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, k) = Product_{j=1..n} (q*(3*q - 1)/2)^j and q = 3.
T(n, k, q) = (binomial(3*q, 2)/3)^(k*(n-k)) with q = 3.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 10. - G. C. Greubel, Jun 30 2021

Extensions

Edited by G. C. Greubel, Jun 30 2021

A176639 Triangle T(n, k) = 15^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 15, 1, 1, 225, 225, 1, 1, 3375, 50625, 3375, 1, 1, 50625, 11390625, 11390625, 50625, 1, 1, 759375, 2562890625, 38443359375, 2562890625, 759375, 1, 1, 11390625, 576650390625, 129746337890625, 129746337890625, 576650390625, 11390625, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 22 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     15,          1;
  1,    225,        225,           1;
  1,   3375,      50625,        3375,          1;
  1,  50625,   11390625,    11390625,      50625,      1;
  1, 759375, 2562890625, 38443359375, 2562890625, 759375, 1;

Links

Crossrefs

Cf. A000384.
Cf. A158116 (q=2), this sequence (q=3), A176641 (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), this sequence (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).

Programs

  • Magma
    [(15)^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 30 2021
  • Mathematica
    (* First program *)
T[n_, k_, q_] = Binomial[2*q, 2]^(k*(n-k));
Table[T[n, k, 3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 30 2021 *)
(* Second program *)
With[{m=13}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)
  • Sage
    flatten([[(15)^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

Formula

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, k) = Product_{j=1..n} (q*(2*q - 1))^j and q = 3.
T(n, k, q) = binomial(2*q, 2)^(k*(n-k)) with q = 3.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 13. - G. C. Greubel, Jun 30 2021

Extensions

Edited by G. C. Greubel, Jun 30 2021

A176642 Triangle T(n, k) = 8^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 64, 64, 1, 1, 512, 4096, 512, 1, 1, 4096, 262144, 262144, 4096, 1, 1, 32768, 16777216, 134217728, 16777216, 32768, 1, 1, 262144, 1073741824, 68719476736, 68719476736, 1073741824, 262144, 1, 1, 2097152, 68719476736, 35184372088832, 281474976710656, 35184372088832, 68719476736, 2097152, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 22 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,      1;
  1,      8,          1;
  1,     64,         64,           1;
  1,    512,       4096,         512,           1;
  1,   4096,     262144,      262144,        4096,          1;
  1,  32768,   16777216,   134217728,    16777216,      32768,      1;
  1, 262144, 1073741824, 68719476736, 68719476736, 1073741824, 262144, 1;

Links

Crossrefs

Cf. this sequence (q=2), A176643 (q=3), A176644 (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), this sequence (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).
Cf. A000567, A004247.

Programs

  • Magma
    [8^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 30 2021
  • Mathematica
    T[n_, k_, q_]:= (q*(3*q-2))^(k*(n-k)); Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten
With[{m=6}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)
  • Sage
    flatten([[8^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

Formula

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, q) = (q*(3*q - 2))^binomial(n+1,2) and q = 2.
T(n, k, q) = (q*(3*q-2))^(k*(n-k)) with q = 2.
T(n, k) = 8^A004247(n,k), where A004247 is interpreted as a triangle. [relation detected by sequencedb.net]. - R. J. Mathar, Jun 30 2021
T(n, k, m) = (m+2)^(k*(n-k)) with m = 6. - G. C. Greubel, Jun 30 2021

Extensions

Edited by R. J. Mathar and G. C. Greubel, Jun 30 2021

A176631 Triangle T(n, k) = 22^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 22, 1, 1, 484, 484, 1, 1, 10648, 234256, 10648, 1, 1, 234256, 113379904, 113379904, 234256, 1, 1, 5153632, 54875873536, 1207269217792, 54875873536, 5153632, 1, 1, 113379904, 26559922791424, 12855002631049216, 12855002631049216, 26559922791424, 113379904, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 22 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      22,           1;
  1,     484,         484,             1;
  1,   10648,      234256,         10648,           1;
  1,  234256,   113379904,     113379904,      234256,       1;
  1, 5153632, 54875873536, 1207269217792, 54875873536, 5153632, 1;

Links

Crossrefs

Cf. A000326.
Cf. A118190 (q=2), A176627 (q=3), this sequence (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), this sequence (m=20), A176641 (m=26), A176644 (m=38).

Programs

  • Magma
    [22^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 01 2021
  • Mathematica
    T[n_, k_, q_]= (Binomial[3*q,2]/3)^(k*(n-k)); Table[T[n,k,4], {n,0,12}, {k,0,n}]//Flatten
Table[22^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 01 2021 *)
  • Sage
    flatten([[22^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 01 2021

Formula

T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, k) = Product_{j=1..n} (q*(3*q - 1)/2)^j and q = 4.
T(n, k, q) = (binomial(3*q, 2)/3)^(k*(n-k)) with q = 4.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 20. - G. C. Greubel, Jul 01 2021

Extensions

Edited by G. C. Greubel, Jul 01 2021

A176641 Triangle T(n, k) = 28^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 28, 1, 1, 784, 784, 1, 1, 21952, 614656, 21952, 1, 1, 614656, 481890304, 481890304, 614656, 1, 1, 17210368, 377801998336, 10578455953408, 377801998336, 17210368, 1, 1, 481890304, 296196766695424, 232218265089212416, 232218265089212416, 296196766695424, 481890304, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 22 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,        1;
  1,       28,            1;
  1,      784,          784,              1;
  1,    21952,       614656,          21952,            1;
  1,   614656,    481890304,      481890304,       614656,        1;
  1, 17210368, 377801998336, 10578455953408, 377801998336, 17210368, 1;

Links

Crossrefs

Cf. A000384.
Cf. A158116 (q=2), A176639 (q=3), this sequence (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), this sequence (m=26).
Cf. A007318 (p=0), A118180 (p=1), A158116 (p=2), A158117 (p=3), A176639 (p=4), A176643 (p=5), this sequence (p=6).

Programs

  • Magma
    [(28)^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 30 2021
  • Mathematica
    T[n_, k_, q_] = Binomial[2*q, 2]^(k*(n-k));
Table[T[n, k, 4], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 30 2021 *)
With[{m=26}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)
  • Sage
    flatten([[(28)^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

Formula

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, k) = Product_{j=1..n} (q*(2*q - 1))^j and q = 4.
From G. C. Greubel, Jun 30 2021: (Start)
T(n, k, q) = binomial(2*q, 2)^(k*(n-k)) with q = 4.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 26.
T(n, k, p) = binomial(p+2, 2)^(k*(n-k)) with p = 6. (End)

Extensions

Edited by G. C. Greubel, Jun 30 2021

A176643 Triangle T(n, k) = 21^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 21, 1, 1, 441, 441, 1, 1, 9261, 194481, 9261, 1, 1, 194481, 85766121, 85766121, 194481, 1, 1, 4084101, 37822859361, 794280046581, 37822859361, 4084101, 1, 1, 85766121, 16679880978201, 7355827511386641, 7355827511386641, 16679880978201, 85766121, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 22 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      21,           1;
  1,     441,         441,            1;
  1,    9261,      194481,         9261,           1;
  1,  194481,    85766121,     85766121,      194481,       1;
  1, 4084101, 37822859361, 794280046581, 37822859361, 4084101, 1;

Links

Crossrefs

Cf. A000567.
Cf. A176642 (q=2), this sequence (q=3), A176644 (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), this sequence (m=19), A176631 (m=20), A176641 (m=26).

Programs

  • Magma
    [(21)^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 01 2021
  • Mathematica
    T[n_, k_, q_]:= (q*(3*q-2))^(k*(n-k)); Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten
Table[21^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 01 2021 *)
  • Sage
    flatten([[(21)^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 01 2021

Formula

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, q) = (q*(3*q - 2))^binomial(n+1,2) and q = 3.
T(n, k, q) = (q*(3*q-2))^(k*(n-k)) with q = 3.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 19. - G. C. Greubel, Jul 01 2021

Extensions

Edited by G. C. Greubel, Jul 01 2021

A176644 Triangle T(n, k) = 40^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 40, 1, 1, 1600, 1600, 1, 1, 64000, 2560000, 64000, 1, 1, 2560000, 4096000000, 4096000000, 2560000, 1, 1, 102400000, 6553600000000, 262144000000000, 6553600000000, 102400000, 1, 1, 4096000000, 10485760000000000, 16777216000000000000, 16777216000000000000, 10485760000000000, 4096000000, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 22 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,         1;
  1,        40,             1;
  1,      1600,          1600,               1;
  1,     64000,       2560000,           64000,             1;
  1,   2560000,    4096000000,      4096000000,       2560000,         1;
  1, 102400000, 6553600000000, 262144000000000, 6553600000000, 102400000, 1;

Links

Crossrefs

Cf. A000567.
Cf. A176642 (q=2), A176643 (q=3), this sequence (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26), this sequence (m=38).

Programs

  • Magma
    [40^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 01 2021
  • Mathematica
    T[n_, k_, q_]:= (q*(3*q-2))^(k*(n-k)); Table[T[n, k, 4], {n, 0, 12}, {k, 0, n}]//Flatten
Table[(40)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 01 2021 *)
  • Sage
    flatten([[40^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 01 2021

Formula

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, q) = (q*(3*q - 2))^binomial(n+1,2) and q = 4.
T(n, k, q) = (q*(3*q-2))^(k*(n-k)) with q = 4.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 38. - G. C. Greubel, Jul 01 2021

Extensions

Edited by G. C. Greubel, Jul 01 2021

A378666 Triangular array read by rows: T(n,k) is the number of n X n idempotent matrices over GF(3) having rank k, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 117, 117, 1, 1, 1080, 10530, 1080, 1, 1, 9801, 882090, 882090, 9801, 1, 1, 88452, 72243171, 666860040, 72243171, 88452, 1, 1, 796797, 5873190687, 491992666011, 491992666011, 5873190687, 796797, 1, 1, 7173360, 476309310660, 360089838858960, 3267815287645062, 360089838858960, 476309310660, 7173360, 1
Offset: 0

Views

Author

Geoffrey Critzer, Dec 02 2024

Keywords

Comments

A matrix M is idempotent if M^2 = M.

Examples

			Triangle T(n,k) begins:
  1;
  1,    1;
  1,   12,      1;
  1,  117,    117,      1;
  1, 1080,  10530,   1080,    1;
  1, 9801, 882090, 882090, 9801, 1;
  ...

Crossrefs

Cf. A296548, A053846 (row sums).
Cf. A022167, A118180.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, b(n-1, k-1)+3^k*b(n-1, k)))
    end:
T:= (n,k)-> 3^(k*(n-k))*b(n, k):
seq(seq(T(n,k), k=0..n), n=0..8);  # Alois P. Heinz, Dec 02 2024
  • Mathematica
    nn = 8; \[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}]; B[n_, q_] := \[Gamma][n, q]/(q - 1)^n; \[Zeta][x_] := Sum[x^n/B[n, 3], {n, 0, nn}];Map[Select[#, # > 0 &] &, Table[B[n, 3], {n,0,nn}]*CoefficientList[Series[\[Zeta][x] \[Zeta][y x], {x, 0, nn}], {x, y}]] // Flatten

Formula

Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/B(n) = e(x)*e(y*x) where e(x) = Sum_{n>=0} x^n/B(n) and B(n) = A053290(n)/2^n.
T(n,k) = A022167(n,k) * A118180(n,k). - Alois P. Heinz, Dec 02 2024

A156582 Square array T(n, k) = (k+2)^binomial(n, 2) with T(n, 0) = n!, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 27, 24, 1, 1, 5, 64, 729, 120, 1, 1, 6, 125, 4096, 59049, 720, 1, 1, 7, 216, 15625, 1048576, 14348907, 5040, 1, 1, 8, 343, 46656, 9765625, 1073741824, 10460353203, 40320, 1, 1, 9, 512, 117649, 60466176, 30517578125, 4398046511104, 22876792454961, 362880
Offset: 0

Views

Author

Roger L. Bagula, Feb 10 2009

Keywords

Examples

			Square array begins as:
    1,     1,       1,       1,        1,         1 ...;
    1,     1,       1,       1,        1,         1 ...;
    2,     3,       4,       5,        6,         7 ...;
    6,    27,      64,     125,      216,       343 ...;
   24,   729,    4096,   15625,    46656,    117649 ...;
  120, 59049, 1048576, 9765625, 60466176, 282475249 ...;
Antidiagonal triangle begins as:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 3,   6;
  1, 1, 4,  27,    24;
  1, 1, 5,  64,   729,     120;
  1, 1, 6, 125,  4096,   59049,        720;
  1, 1, 7, 216, 15625, 1048576,   14348907,        5040;
  1, 1, 8, 343, 46656, 9765625, 1073741824, 10460353203, 40320;

Links

Crossrefs

Cf. A118180, A118185, A118190.

Programs

  • Magma
    A156582:= func< n,k | k eq 0 select Factorial(n) else (k+2)^Binomial(n,2) >;
[A156582(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 28 2021
  • Mathematica
    (* First program *)
T[n_, k_]:= If[k==0, n!, Product[Sum[Binomial[j-1,i]*(k+1)^i, {i,0,j-1}], {j,n}]];
Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 28 2021 *)
(* Second program *)
T[n_, k_]:= If[k==0, n!, (k+2)^Binomial[n, 2]];
Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 28 2021 *)
  • Sage
    def A156582(n,k): return factorial(n) if (k==0) else (k+2)^binomial(n,2)
flatten([[A156582(k,n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 28 2021

Formula

T(n,k) = Product_{j=1..n} ( Sum_{i=0..j-1} binomial(j-1, i)*(k+1)^i ) with T(n, 0) = n! (square array).
T(n, k) = (k+2)^binomial(n, 2) with T(n, 0) = n! (square array). - G. C. Greubel, Jun 28 2021

Extensions

Edited by G. C. Greubel, Jun 28 2021
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