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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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

A069778 q-factorial numbers 3!_q.

Original entry on oeis.org

1, 6, 21, 52, 105, 186, 301, 456, 657, 910, 1221, 1596, 2041, 2562, 3165, 3856, 4641, 5526, 6517, 7620, 8841, 10186, 11661, 13272, 15025, 16926, 18981, 21196, 23577, 26130, 28861, 31776, 34881, 38182, 41685, 45396, 49321, 53466, 57837, 62440, 67281, 72366
Offset: 0

Views

Author

Franklin T. Adams-Watters, Apr 07 2002

Keywords

Comments

Number of proper n-colorings of the 4-cycle with one vertex color fixed (offset 2). - Michael Somos, Jul 19 2002
n such that x^3 + x^2 + x + n factors over the integers. - James R. Buddenhagen, Apr 19 2005
If Y is a 4-subset of an n-set X then, for n>=5, a(n-5) is the number of 5-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 08 2007
Equals row sums of the Connell (A001614) sequence read as a triangle. - Gary W. Adamson, Sep 01 2008
Binomial transform of 1, 5, 10, 6, 0, 0, 0 (0 continued). - Philippe Deléham, Mar 17 2014
Digital root is A251780. - Peter M. Chema, Jul 11 2016

Examples

			For 2-colorings only 1212 is proper so a(2-2)=1. The proper 3-colorings are: 1212,1313,1213,1312,1232,1323 so a(3-2)=6.
a(0) = 1*1 = 1;
a(1) = 1*1 + 5*1 = 6;
a(2) = 1*1 + 5*2 + 10*1 = 21;
a(3) = 1*1 + 5*3 + 10*3 + 6*1 = 52;
a(4) = 1*1 + 5*4 + 10*6 + 6*4 = 105; etc. - _Philippe Deléham_, Mar 17 2014

References

  • T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

Links

Crossrefs

Cf. A069777, A069779, A218503, A056108 (first differences).
Cf. A001614. - Gary W. Adamson, Sep 01 2008
Cf. A226449. - Bruno Berselli, Jun 09 2013

Programs

  • Maple
    A069778 := proc(n)
    (n+1)*(n^2+n+1) ;
end proc: # R. J. Mathar, Aug 24 2013
  • Mathematica
    LinearRecurrence[{4, -6, 4, -1}, {1, 6, 21, 52}, 41] (* or *) Table[(n + 1) (n^2 + n + 1), {n, 0, 41}] (* Harvey P. Dale, Jul 11 2011 *)
Table[QFactorial[3, n], {n, 0, 41}] (* Arkadiusz Wesolowski, Oct 31 2012 *)
  • PARI
    a(n)=(n+1)*(n^2+n+1)

Formula

a(n) = (n + 1)*(n^2 + n + 1).
a(n) = (n+1)^3-2*T(n) where T(n) =n*(n+1)/2= A000217(n) is the n-th triangular number. - Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 14 2006
a(n) = n^8 mod (n^3+n), with offset 1..a(1)=1. - Gary Detlefs, May 02 2010
a(n) = 4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4), n>3. - Harvey P. Dale, Jul 11 2011
G.f.: (1+2*x+3*x^2)/(1-x)^4. - Harvey P. Dale, Jul 11 2011
For n>0 a(n) = Sum_{k=A000217(n-1)...A000217(n+1)} k. - J. M. Bergot, Feb 11 2015
E.g.f.: (1 + 5*x + 5*x^2 + x^3)*exp(x). - Ilya Gutkovskiy, Jul 11 2016

A218503 q-factorial numbers 5!_q.

Original entry on oeis.org

1, 120, 9765, 251680, 3043425, 22661496, 121226245, 510902400, 1799118945, 5507702200, 15072415941, 37630041120, 87029433985, 188664603960, 386925380325, 756298318336, 1417430759745, 2559798038520, 4472991338725, 7589075296800, 12538953723681
Offset: 0

Views

Author

Arkadiusz Wesolowski, Oct 31 2012

Keywords

Links

Crossrefs

Cf. A069777, A069778, A069779.

Programs

  • Mathematica
    Table[QFactorial[5, n], {n, 0, 20}]
Join[{1},With[{f=Times@@Table[Total[n^Range[0,i]],{i,4}]},Table[f,{n,20}]]] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,120,9765,251680,3043425,22661496,121226245,510902400,1799118945,5507702200,15072415941},30] (* Harvey P. Dale, Sep 04 2017 *)

Formula

a(n) = (n + 1)*(n^2 + n + 1)*(n^3 + n^2 + n + 1)*(n^4 + n^3 + n^2 + n + 1).
G.f.: (1 + x*(109 + x*(8500 + x*(150700 + x*(792550 + x*(1454134 + x*(978436 + 5*x*(45788 + x*(3053 + 33*x)))))))))/(1 - x)^11.
Showing 1-3 of 3 results.

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