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%I #28 Feb 14 2024 09:08:58
%S 1,120,9765,251680,3043425,22661496,121226245,510902400,1799118945,
%T 5507702200,15072415941,37630041120,87029433985,188664603960,
%U 386925380325,756298318336,1417430759745,2559798038520,4472991338725,7589075296800,12538953723681
%N q-factorial numbers 5!_q.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F a(n) = (n + 1)*(n^2 + n + 1)*(n^3 + n^2 + n + 1)*(n^4 + n^3 + n^2 + n + 1).
%F G.f.: (1 + x*(109 + x*(8500 + x*(150700 + x*(792550 + x*(1454134 + x*(978436 + 5*x*(45788 + x*(3053 + 33*x)))))))))/(1 - x)^11.
%t Table[QFactorial[5, n], {n, 0, 20}]
%t Join[{1},With[{f=Times@@Table[Total[n^Range[0,i]],{i,4}]},Table[f,{n,20}]]] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,120,9765,251680,3043425,22661496,121226245,510902400,1799118945,5507702200,15072415941},30] (* _Harvey P. Dale_, Sep 04 2017 *)
%Y Cf. A069777, A069778, A069779.
%K easy,nonn
%O 0,2
%A _Arkadiusz Wesolowski_, Oct 31 2012