|
%I #37 Jan 23 2022 13:48:09
%S 1,2,1,6,3,1,30,15,5,1,210,105,35,7,1,2310,1155,385,77,11,1,30030,
%T 15015,5005,1001,143,13,1,510510,255255,85085,17017,2431,221,17,1,
%U 9699690,4849845,1616615,323323,46189,4199,323,19,1,223092870,111546435,37182145,7436429,1062347,96577,7429,437,23,1
%N Triangle read by rows: T(n,k) = prime(n)#/prime(k)#, 0<=k<=n.
%C T(n,k) is the (k+1)-th product of (n-k) successive primes (k, n-(k+1) >= 0). - _Alois P. Heinz_, Jan 21 2022
%H Alois P. Heinz, <a href="/A096334/b096334.txt">Rows n = 0..140, flattened</a>
%F T(n,0) = A002110(n); T(n,n) = 1;
%F T(n,n-1) = A000040(n) for n>0;
%F T(n,k) = A002110(n)/A002110(k), 0<=k<=n.
%F T(n,k) = Product_{j=k+1..n} prime(j). - _Alois P. Heinz_, Jan 21 2022
%e Triangle begins:
%e 1;
%e 2, 1;
%e 6, 3, 1;
%e 30, 15, 5, 1;
%e 210, 105, 35, 7, 1;
%e ...
%p T:= proc(n, k) option remember;
%p `if`(n=k, 1, T(n-1, k)*ithprime(n))
%p end:
%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Jan 21 2022
%t T[n_, k_] := Times @@ Prime[Range[k + 1, n]];
%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 13 2021 *)
%o (PARI) pr(n) = factorback(primes(n)); \\ A002110
%o row(n) = my(P=pr(n)); vector(n+1, k, P/pr(k-1)); \\ _Michel Marcus_, Jan 21 2022
%Y T(n-1+j,n-1) give (j=1-12): A000040, A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127342, A127343, A127344.
%Y Columns k=0-1 give: A002110, A070826.
%Y T(2n,n) gives A107712.
%Y Row sums give A350895.
%Y Antidiagonal sums give A350758.
%Y Cf. A073485 (distinct values sorted).
%K nonn,tabl
%O 0,2
%A _Reinhard Zumkeller_, Aug 03 2004
|
