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

A060757 a(n) = 4^(n^2).

Original entry on oeis.org

1, 4, 256, 262144, 4294967296, 1125899906842624, 4722366482869645213696, 316912650057057350374175801344, 340282366920938463463374607431768211456, 5846006549323611672814739330865132078623730171904, 1606938044258990275541962092341162602522202993782792835301376
Offset: 0

Views

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 23 2001

Keywords

Comments

Number of n X n matrices over GF(4).
a(n) = k^(n^2) with k = 2, 3, 4,... counts n X n matrices over GF(k). - Vincenzo Origlio (vincenzo.origlio(AT)itc.cnr.it), Nov 14 2002

Links

Crossrefs

Cf. A060761.
Cf. A060722.
Cf. A118185.

Programs

Formula

a(n) = A118185(2n,n). - Alois P. Heinz, Jun 29 2021

Extensions

More terms from Philippe Deléham, Nov 19 2007

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

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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