A266743 - OEIS

%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