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A069777 Array of q-factorial numbers n!_q, read by ascending antidiagonals.

%I A069777 #52 Feb 28 2025 11:38:50
%S A069777 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,
%T A069777 9765,2080,105,6,1,1,1,5040,615195,251680,8925,186,7,1,1,1,40320,
%U A069777 78129765,91611520,3043425,29016,301,8,1,1
%N A069777 Array of q-factorial numbers n!_q, read by ascending antidiagonals.
%H A069777 Alois P. Heinz, <a href="/A069777/b069777.txt">Antidiagonals n = 0..55, flattened</a>
%H A069777 Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%H A069777 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A069777 T(n,q) = Product_{k=1..n} (q^k - 1) / (q - 1).
%F A069777 T(n,k) = Product_{n1=k..n-1} A104878(n1,k). - _Johannes W. Meijer_, Aug 21 2011
%F A069777 T(n,k) = Sum_{i>=0} A008302(n,i)*k^i. - _Geoffrey Critzer_, Feb 26 2025
%e A069777 Square array begins:
%e A069777     1,   1,    1,      1,       1,        1,         1, ...
%e A069777     1,   1,    1,      1,       1,        1,         1, ...
%e A069777     1,   2,    3,      4,       5,        6,         7, ...
%e A069777     1,   6,   21,     52,     105,      186,       301, ...
%e A069777     1,  24,  315,   2080,    8925,    29016,     77959, ...
%e A069777     1, 120, 9765, 251680, 3043425, 22661496, 121226245, ...
%e A069777     ...
%p A069777 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
%p A069777 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
%p A069777 # alternative Maple program:
%p A069777 T:= proc(n, k) option remember; `if`(n<2, 1,
%p A069777       T(n-1, k)*`if`(k=1, n, (k^n-1)/(k-1)))
%p A069777     end:
%p A069777 seq(seq(T(d-k, k), k=0..d), d=0..10);  # _Alois P. Heinz_, Sep 08 2021
%t A069777 (* Returns the rectangular array *) Table[Table[QFactorial[n, q], {q, 0, 6}], {n, 0, 6}] (* _Geoffrey Critzer_, May 21 2017 *)
%o A069777 (PARI) T(n,q)=prod(k=1, n, ((q^k - 1) / (q - 1))) \\ _Andrew Howroyd_, Feb 19 2018
%Y A069777 Columns q=0..11 are A000012, A000142, A005329, A015001, A015002, A015004, A015005, A015006, A015007, A015008, A015009, A015011.
%Y A069777 Rows n=3..5 are A069778, A069779, A218503.
%Y A069777 Main diagonal gives A347611.
%Y A069777 Cf. A008302, A156173.
%K A069777 easy,nonn,tabl
%O A069777 0,8
%A A069777 _Franklin T. Adams-Watters_, Apr 07 2002
%E A069777 Name edited by _Michel Marcus_, Sep 08 2021