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A069777 Triangle of q-factorial numbers n!_q, for (n,q) = (0,0), (1,0), (0,1), (2,0), etc. 9
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

Table of n, a(n) for n=0..54.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

FORMULA

T(n,q) = Prod_{k=1..n} (q^k - 1) / (q - 1).

T(n,k) = Prod_{n1=k..n-1} A104878(n1,k). - Johannes W. Meijer, Aug 21 2011

EXAMPLE

    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

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

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

Columns q=1..11 are A000142, A005329, A015001, A015002, A015004, A015005, A015006, A015007, A015008, A015009, A015011.

Rows n=3..5 are A069778, A069779, A218503.

Cf. A156173.

Sequence in context: A332700 A256268 A213275 * A225816 A227655 A064992

Adjacent sequences:  A069774 A069775 A069776 * A069778 A069779 A069780

KEYWORD

easy,nonn,tabl

AUTHOR

Franklin T. Adams-Watters, Apr 07 2002

STATUS

approved

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Last modified April 16 00:05 EDT 2021. Contains 343020 sequences. (Running on oeis4.)