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A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

%I A309131 #54 Sep 08 2022 08:46:21
%S A309131 2,4,3,6,6,5,8,9,10,7,10,12,15,14,11,12,15,20,21,22,13,14,18,25,28,33,
%T A309131 26,17,16,21,30,35,44,39,34,19,18,24,35,42,55,52,51,38,23,20,27,40,49,
%U A309131 66,65,68,57,46,29,22,30,45,56,77,78,85,76,69,58,31
%N A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.
%C A309131 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).
%C A309131 The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
%C A309131 The k-th column of the triangle T gives the multiples of prime(1 + k).
%C A309131 Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - _Michel Marcus_, Jul 15 2019
%H A309131 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz matrix</a>
%F A309131 T(n, k) = A025581(n, k)*A000040(1 + k).
%e A309131 The triangle T(n, k) begins:
%e A309131 ---+-----------------------------------------------------
%e A309131 n\k|    0     1     2     3     4     5     6     7     8
%e A309131 ---+-----------------------------------------------------
%e A309131 1  |    2
%e A309131 2  |    4     3
%e A309131 3  |    6     6     5
%e A309131 4  |    8     9    10     7
%e A309131 5  |   10    12    15    14    11
%e A309131 6  |   12    15    20    21    22    13
%e A309131 7  |   14    18    25    28    33    26    17
%e A309131 8  |   16    21    30    35    44    39    34    19
%e A309131 9  |   18    24    35    42    55    52    51    38    23
%e A309131 ...
%e A309131 For n = 3 the matrix M(3) is
%e A309131           2,         3,         5
%e A309131     M_{2,1},         2,         3
%e A309131     M_{3,1},   M_{3,2},         2
%e A309131 and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.
%p A309131 a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);
%t A309131 Flatten[Table[(n-k)*Prime[1+k],{n,1,11},{k,0,n-1}]]
%o A309131 (Magma) [[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output
%o A309131 (PARI)
%o A309131 T(n, k) = (n - k)*prime(1 + k);
%o A309131 tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output
%o A309131 (Sage) [[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output
%Y A309131 Cf. A025581, A318173, A325516.
%Y A309131 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.
%K A309131 nonn,easy,tabl
%O A309131 1,1
%A A309131 _Stefano Spezia_, Jul 14 2019