cp's OEIS Frontend

A364812 Triangle of generalized binomial coefficients T(n,k) = ff(n)/(ff(k)*ff(n-k)) where ff(n) = A363838(n), the generalized factorial.

%I A364812 #12 Oct 22 2023 01:25:32
%S A364812 1,1,1,1,2,1,1,3,3,1,1,16,24,16,1,1,5,40,40,5,1,1,36,90,480,90,36,1,1,
%T A364812 7,126,210,210,126,7,1,1,256,896,10752,3360,10752,896,256,1,1,81,
%U A364812 10368,24192,54432,54432,24192,10368,81,1,1,100,4050,345600,151200,1088640,151200,345600,4050,100,1
%N A364812 Triangle of generalized binomial coefficients T(n,k) = ff(n)/(ff(k)*ff(n-k)) where ff(n) = A363838(n), the generalized factorial.
%H A364812 Michel Marcus, <a href="/A364812/b364812.txt">Table of n, a(n) for n = 0..5150</a> (Rows n=0..100 flattened).
%H A364812 Jeffrey C. Lagarias and Wijit Yangjit, <a href="https://arxiv.org/abs/2310.12949">The factorial function and generalizations, extended</a>, arXiv:2310.12949 [math.NT], 2023. See Table 3 p. 30.
%F A364812 T(n,k) = A363838(n)/(A363838(k)*A363838(n-k)).
%e A364812 Triangle begins:
%e A364812   1;
%e A364812   1,  1;
%e A364812   1,  2,  1;
%e A364812   1,  3,  3,   1;
%e A364812   1, 16, 24,  16,  1;
%e A364812   1,  5, 40,  40,  5, 1;
%e A364812   1, 36, 90, 480, 90, 36, 1;
%e A364812   ...
%o A364812 (PARI)
%o A364812 f(n, b) = sum(i=1, logint(n, b), n\b^i);
%o A364812 ff(n) = prod(b=2, n, b^f(n,b)); \\ A363838
%o A364812 T(n,k) = ff(n)/(ff(k)*ff(n-k));
%o A364812 row(n) = vector(n+1, k, T(n, k-1));
%Y A364812 Cf. A007318, A363838.
%K A364812 nonn,tabl
%O A364812 0,5
%A A364812 _Michel Marcus_, Oct 21 2023