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A069777
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Triangle of q-factorial numbers n!_q, for (n,q) = (0,0), (1,0), (0,1), (2,0), etc.
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9
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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
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OFFSET
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0,8
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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(*Returns the rectangular array*) Table[Table[QFactorial[n, q], {q, 0, 6}], {n, 0, 6}] (* Geoffrey Critzer, May 21 2017 *)
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PROG
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(PARI) T(n, q)=prod(k=1, n, ((q^k - 1) / (q - 1))) \\ Andrew Howroyd, Feb 19 2018
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CROSSREFS
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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
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Franklin T. Adams-Watters, Apr 07 2002
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STATUS
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approved
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