A069777 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

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

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

EXAMPLE

Square array begins:

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, ...

...

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 *)

PROG

(PARI) T(n, q)=prod(k=1, n, ((q^k - 1) / (q - 1))) \\ Andrew Howroyd, Feb 19 2018

CROSSREFS