G. C. Greubel, <a href= "/A176271/b176271_1.txt ">Rows n = 1..100 of the triangle, flattened</a>

reviewed

approved

proposed

reviewed

editing

proposed

Sum_{j=1..n} (Sum_{k=1..jn} T(n,j,k)) =A000537(n) (sum of first n rows).

proposed

editing

editing

proposed

T(n,k) =A005408(n*(n-1)/2 + k - 1);

Nicomachus: row sums giveA000578;

A000537(n) = sum of first n rows;

ABS(alternating row sums) giveA065599;

central terms giveA016754:T(2*n-1,n) =A016754(n-1);

T(2*n,n) =A000466(n); T(2*n,n+1) =A053755(n);

T(n,k) + T(n,n-k+1) =A001105(n), 1 <= k <= n;

T(n,1) =A002061(n), central polygonal numbers;

T(n,2) =A027688(n-1) for n > 1;

T(n,3) =A027690(n-1) for n > 2;

T(n,4) =A027692(n-1) for n > 3;

T(n,5) =A027694(n-1) for n > 4;

T(n,6) =A048058(n-1) for n > 5;

T(n,n-3) =A108195(n-2) for n > 3;

T(n,n-2) =A082111(n-2) for n > 2;

T(n,n-1) =A014209(n-1) for n > 1;

T(n,n) =A028387(n-1);

G. C. Greubel, <a href= "/A176271/b176271_1.txt ">Rows n = 1..100 of the triangle, flattened</a>

T(n,k) = n^2 - n + 2*k - 1,for1 <= k <= n.

T(n, k) =A005408(n*(n-1)/2 + k - 1).

T(2*n-1, n) =A016754(n-1) (main diagonal).

T(2*n, n) =A000466(n).

T(2*n, n+1) =A053755(n).

T(n, k) + T(n, n-k+1) =A001105(n), 1 <= k <= n.

T(n, 1) =A002061(n), central polygonal numbers.

T(n, 2) =A027688(n-1) for n > 1.

T(n, 3) =A027690(n-1) for n > 2.

T(n, 4) =A027692(n-1) for n > 3.

T(n, 5) =A027694(n-1) for n > 4.

T(n, 6) =A048058(n-1) for n > 5.

T(n, n-3) =A108195(n-2) for n > 3.

T(n, n-2) =A082111(n-2) for n > 2.

T(n, n-1) =A014209(n-1) for n > 1.

T(n, n) =A028387(n-1).

Sum_{k=1..n} T(n, k) =A000578(n) (Nicomachus's theorem).

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A065599(n) (alternating sign row sums).

Sum_{j=1..n} (Sum_{k=1..j} T(n, k)) =A000537(n) (sum of first n rows).

Table[n^2-n+2*k-1, {n, 15}, {k, n}]//Flatten (*G. C. Greubel,Mar 10 2024 *)

(Magma) [n^2-n+2*k-1: k in [1..n], n in [1..15]]; //G. C. Greubel,Mar 10 2024

(SageMath) flatten([[n^2-n+2*k-1 for k in range(1, n+1)] for n in range(1, 16)]) #G. C. Greubel,Mar 10 2024

Cf.A214604,A214661A000466,A000537,A000578,A001105,A002061,A005408,A014209.

Cf.A016754,A027688,A027690,A027692,A027694,A028387,A048058.

Cf.A053755,A065599,A082111,A108195,A108309,A214604,A214661.

approved

editing

proposed

approved

editing

proposed

Wikipedia, <a href= "http://de.wikipedia.org/wiki/Nikomachos_von_Gerasa" >Nikomachos von Gerasa</a>

Wikipedia, <a href= "http://de.wikipedia.org/wiki/Nikomachos_von_Gerasa" >Nikomachos von Gerasa</a>

approved

editing