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 A069777 Array of q-factorial numbers n!_q, read by ascending antidiagonals. 19
 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, 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 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. 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 EXTENSIONS Name edited by Michel Marcus, Sep 08 2021 STATUS approved

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Last modified September 15 13:44 EDT 2024. Contains 375938 sequences. (Running on oeis4.)