|
%I #19 Feb 24 2022 02:04:06
%S 1,1,1,2,3,1,1,2,1,6,15,10,1,2,6,5,1,12,42,42,14,1,3,12,14,7,1,10,45,
%T 60,42,10,1,2,10,15,14,5,1,12,66,110,132,66,22,1,2,12,22,33,22,11,1,
%U 420,2730,5460,10010,8580,6006,910,1
%N Irregular triangle T(n,k) read by rows: see Comments for definition.
%C Let p_i denote the i-th prime, let pi(n) = A000720(n), and let N! = Product_{i = 1..pi(N)} (p_i)^U(N,i) be the prime factorization of N!, where U(N,i) = A115627(N,i).
%C Let V(n,i) = floor(n/(prime(i)-1)) = A266742(n,i).
%C The present triangle is defined by T(n,k) =
%C Product_{i} (p_i)^V(n,i) / ( Product_{j} (p_j)^V(k,j) * Product_{r} (p_r)^U(n-k+1,r) ).
%H H. T. Davis, <a href="/A002443/a002443.pdf">Tables of the Mathematical Functions</a>, Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX. [Annotated scan of pages 204-208 of Volume 2.] See Table 3 on page 207.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 3, 1;
%e 1, 2, 1;
%e 6, 15, 10, 1;
%e 2, 6, 5, 1;
%e 12, 42, 42, 14, 1;
%e 3, 12, 14, 7, 1;
%e 10, 45, 60, 42, 10, 1;
%e 2, 10, 15, 14, 5, 1;
%e 12, 66, 110, 132, 66, 22, 1;
%e 2, 12, 22, 33, 22, 11, 1;
%e 420, 2730, 5460, 10010, 8580, 6006, 910, 1;
%e ...
%Y Cf. A000720, A002443, A002444, A115627, A266742.
%K nonn,tabf
%O 1,4
%A _N. J. A. Sloane_, Jan 08 2016
|
