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.

Showing 1-3 of 3 results.

A069777 Array of q-factorial numbers n!_q, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 21, 4, 1, 1, 1, 120, 315, 52, 5, 1, 1, 1, 720, 9765, 2080, 105, 6, 1, 1, 1, 5040, 615195, 251680, 8925, 186, 7, 1, 1, 1, 40320, 78129765, 91611520, 3043425, 29016, 301, 8, 1, 1
Offset: 0

Views

Author

Franklin T. Adams-Watters, Apr 07 2002

Keywords

Examples

			Square array begins:
    1,   1,    1,      1,       1,        1,         1, ...
    1,   1,    1,      1,       1,        1,         1, ...
    1,   2,    3,      4,       5,        6,         7, ...
    1,   6,   21,     52,     105,      186,       301, ...
    1,  24,  315,   2080,    8925,    29016,     77959, ...
    1, 120, 9765, 251680, 3043425, 22661496, 121226245, ...
    ...

Links

Crossrefs

Columns q=0..11 are A000012, A000142, A005329, A015001, A015002, A015004, A015005, A015006, A015007, A015008, A015009, A015011.
Rows n=3..5 are A069778, A069779, A218503.
Main diagonal gives A347611.
Cf. A008302, A156173.

Programs

  • Maple
    A069777 := proc(n,k) local n1: mul(A104878(n1,k), n1=k..n-1) end: A104878 := proc(n,k): if k = 0 then 1 elif k=1 then n elif k>=2 then (k^(n-k+1)-1)/(k-1) fi: end: seq(seq(A069777(n,k), k=0..n), n=0..9); # Johannes W. Meijer, Aug 21 2011
nmax:=9: T(0,0):=1: for n from 1 to nmax do T(n,0):=1:  T(n,1):= (n-1)! od: for q from 2 to nmax do for n from 0 to nmax do T(n+q,q) := product((q^k - 1)/(q - 1), k= 1..n) od: od: for n from 0 to nmax do seq(T(n,k), k=0..n) od; seq(seq(T(n,k), k=0..n), n=0..nmax); # Johannes W. Meijer, Aug 21 2011
# alternative Maple program:
T:= proc(n, k) option remember; `if`(n<2, 1,
      T(n-1, k)*`if`(k=1, n, (k^n-1)/(k-1)))
    end:
seq(seq(T(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Sep 08 2021
  • Mathematica
    (* Returns the rectangular array *) Table[Table[QFactorial[n, q], {q, 0, 6}], {n, 0, 6}] (* Geoffrey Critzer, May 21 2017 *)
  • PARI
    T(n,q)=prod(k=1, n, ((q^k - 1) / (q - 1))) \\ Andrew Howroyd, Feb 19 2018

Formula

T(n,q) = Product_{k=1..n} (q^k - 1) / (q - 1).
T(n,k) = Product_{n1=k..n-1} A104878(n1,k). - Johannes W. Meijer, Aug 21 2011
T(n,k) = Sum_{i>=0} A008302(n,i)*k^i. - Geoffrey Critzer, Feb 26 2025

Extensions

Name edited by Michel Marcus, Sep 08 2021

A156286 Triangle T(n, k) = (1/k^n)*Product_{j=1..n} ( (k-1)*(k+1)^j + 1 ), read by rows.

Original entry on oeis.org

1, 1, 10, 1, 140, 1419, 1, 5740, 242649, 3350536, 1, 700280, 165729267, 7853656384, 161827775045, 1, 255602200, 452606628177, 92036999164096, 6040221703554625, 193317016162131576, 1, 279628806800, 4943822199577371, 5392815929021041024, 1352701610289354714125, 132670761753844630766736, 6731905265314349384346775
Offset: 1

Views

Author

Roger L. Bagula, Feb 07 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,     10;
  1,    140,      1419;
  1,   5740,    242649,    3350536;
  1, 700280, 165729267, 7853656384, 161827775045;

Links

Crossrefs

Cf. A156173.

Programs

  • Magma
    A156286:= func< n,k | (&*[(k-1)*(k+1)^j + 1: j in [1..n]])/k^n >;
[A156286(n,k): k in [1..n], n in [1..10]]; // G. C. Greubel, Jan 02 2022
  • Mathematica
    T[n_, k_]:= (1/k^n)*Product[(k-1)*(k+1)^j +1, {j,n}];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Jan 02 2022 *)
  • Sage
    def A156286(n,k): return (1/k^n)*product( (k-1)*(k+1)^j +1 for j in (1..n) )
flatten([[A156286(n,k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Jan 02 2022

Formula

T(n, k) = Product_{j=1..n} ( (k+1)^j - Sum_{i=0..k-1} (k+1)^i ).
T(n, k) = (1/k^n)*Product_{j=1..n} ( (k-1)*(k+1)^j + 1 ). - G. C. Greubel, Jan 02 2022

Extensions

Edited by G. C. Greubel, Jan 02 2022

A156291 A triangle sequence of Cyclotomic products: t(n,m)=Product[Cyclotomic[k, m + 1], {k, 1, n}].

Original entry on oeis.org

1, 3, 8, 21, 104, 315, 105, 1040, 5355, 19344, 3255, 125840, 1826055, 15107664, 86590175, 9765, 880880, 23738715, 317260944, 2684295425, 16476602400, 1240155, 962801840, 129637122615, 6196423497264, 150285647959475, 2261529015616800, 23895819265962735
Offset: 1

Views

Author

Roger L. Bagula, Feb 07 2009

Keywords

Comments

Row sums are: {1, 11, 440, 25844, 103652989, 19502788129, 26313960954200884, 1365438215011770727764, 13180641383420867649481463463, 2480700284650078006965956163001113,...}.

Examples

			{1},
{3, 8},
{21, 104, 315},
{105, 1040, 5355, 19344},
{3255, 125840, 1826055, 15107664, 86590175},
{9765, 880880, 23738715, 317260944, 2684295425, 16476602400},
{1240155, 962801840, 129637122615, 6196423497264, 150285647959475, 2261529015616800, 23895819265962735},

Crossrefs

Cf. A156173.

Programs

  • Mathematica
    Clear[t, n, m, i, k];
t[n_, m_] = Product[Cyclotomic[k, m + 1], {k, 1, n}];
Table[Table[t[n, m], {m, 1, n}], {n, 1, 10}];
Flatten[%]
  • PARI
    T(n,m) = prod(k=1, n, polcyclo(k, m+1)); \\ Michel Marcus, Feb 08 2023

Formula

T(n,m) = Product_{k=1..n} Cyclotomic(k, m + 1).
Showing 1-3 of 3 results.

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