A069777 Array of q-factorial numbers n!_q, read by ascending antidiagonals.
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
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
- Alois P. Heinz, Antidiagonals n = 0..55, flattened
- Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
- Index entries for sequences related to factorial numbers
Crossrefs
Programs
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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
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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