OFFSET
1,1
COMMENTS
T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).
The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
The k-th column of the triangle T gives the multiples of prime(1 + k).
Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019
LINKS
Wikipedia, Toeplitz matrix
EXAMPLE
The triangle T(n, k) begins:
---+-----------------------------------------------------
n\k| 0 1 2 3 4 5 6 7 8
---+-----------------------------------------------------
1 | 2
2 | 4 3
3 | 6 6 5
4 | 8 9 10 7
5 | 10 12 15 14 11
6 | 12 15 20 21 22 13
7 | 14 18 25 28 33 26 17
8 | 16 21 30 35 44 39 34 19
9 | 18 24 35 42 55 52 51 38 23
...
For n = 3 the matrix M(3) is
2, 3, 5
M_{2,1}, 2, 3
M_{3,1}, M_{3,2}, 2
and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.
MAPLE
a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);
MATHEMATICA
Flatten[Table[(n-k)*Prime[1+k], {n, 1, 11}, {k, 0, n-1}]]
PROG
(Magma) [[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output
(PARI)
T(n, k) = (n - k)*prime(1 + k);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output
(Sage) [[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output
CROSSREFS
Cf. A000040: diagonal; A001747: 1st subdiagonal; A001748: 2nd subdiagonal; A001749: 3rd subdiagonal; A001750: 4th subdiagonal; A005843: 0th column; A008585: 1st column; A008587: 2nd column; A008589: 3rd column; A008593: 4th column; A008595: 5th column; A008599: 6th column; A008601: 7th column; A014148: row sums; A138636: 5th subdiagonal; A272470: 6th subdiagonal.
KEYWORD
AUTHOR
Stefano Spezia, Jul 14 2019
STATUS
approved