A364812 - OEIS

%I #12 Oct 22 2023 01:25:32

%S 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 7,126,210,210,126,7,1,1,256,896,10752,3360,10752,896,256,1,1,81,

%U 10368,24192,54432,54432,24192,10368,81,1,1,100,4050,345600,151200,1088640,151200,345600,4050,100,1

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

%H Michel Marcus, <a href="/A364812/b364812.txt">Table of n, a(n) for n = 0..5150</a> (Rows n=0..100 flattened).

%H 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 T(n,k) = A363838(n)/(A363838(k)*A363838(n-k)).

%e Triangle begins:

%e   1;

%e   1,  1;

%e   1,  2,  1;

%e   1,  3,  3,   1;

%e   1, 16, 24,  16,  1;

%e   1,  5, 40,  40,  5, 1;

%e   1, 36, 90, 480, 90, 36, 1;

%e   ...

%o (PARI)

%o f(n, b) = sum(i=1, logint(n, b), n\b^i);

%o ff(n) = prod(b=2, n, b^f(n,b)); \\ A363838

%o T(n,k) = ff(n)/(ff(k)*ff(n-k));

%o row(n) = vector(n+1, k, T(n, k-1));

%Y Cf. A007318, A363838.

%K nonn,tabl

%O 0,5

%A _Michel Marcus_, Oct 21 2023