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Triangle read by rows: T(n,1) = prime(n) (the n-th prime); T(n,k) = 0 for k > n; T(n,k) = T(n-1,k) + T(n-1,k-1) for 2 <= k <= n (1 <= k <= n).
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%I #16 Oct 11 2021 09:18:12

%S 2,3,2,5,5,2,7,10,7,2,11,17,17,9,2,13,28,34,26,11,2,17,41,62,60,37,13,

%T 2,19,58,103,122,97,50,15,2,23,77,161,225,219,147,65,17,2,29,100,238,

%U 386,444,366,212,82,19,2,31,129,338,624,830,810,578,294,101,21,2,37,160,467

%N Triangle read by rows: T(n,1) = prime(n) (the n-th prime); T(n,k) = 0 for k > n; T(n,k) = T(n-1,k) + T(n-1,k-1) for 2 <= k <= n (1 <= k <= n).

%C Sum of row n = A125180(n).

%F T(n,2) = A007504(n-1) (n>=2).

%e Triangle starts:

%e 2;

%e 3, 2;

%e 5, 5, 2;

%e 7, 10, 7, 2;

%e 11, 17, 17, 9, 2;

%e 13, 28, 34, 26, 11, 2;

%e 17, 41, 62, 60, 37, 13, 2;

%p T:=proc(n,k) if k=1 then ithprime(n) elif k>n then 0 else T(n-1,k)+T(n-1,k-1) fi end: for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

%t nmax = 11;

%t row[1] = Prime[Range[nmax]];

%t row[n_] := row[n] = row[n-1] // Accumulate;

%t T[n_, k_] := row[n][[k]];

%t Table[T[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Oct 11 2021 *)

%Y Cf. A125180 (row sums), A007442, A254858 (rows reversed).

%Y Cf. A007504.

%K nonn,tabl

%O 1,1

%A _Gary W. Adamson_, Nov 22 2006

%E Edited by _N. J. A. Sloane_, Dec 02 2006