A126890 Triangle read by rows: T(n,k) = n*(n+2*k+1)/2, 0 <= k <= n.

0, 1, 2, 3, 5, 7, 6, 9, 12, 15, 10, 14, 18, 22, 26, 15, 20, 25, 30, 35, 40, 21, 27, 33, 39, 45, 51, 57, 28, 35, 42, 49, 56, 63, 70, 77, 36, 44, 52, 60, 68, 76, 84, 92, 100, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 55, 65, 75, 85, 95, 105, 115, 125, 135, 145, 155, 66, 77, 88

Offset: 0

Author

Reinhard Zumkeller, Dec 30 2006

Keywords

• nonn
• tabl
• easy

Comments

T(n,k) + T(n,n-k) = A014105(n);

row sums give A059270; Sum_{k=0..n-1} T(n,k) = A000578(n);

central terms give A007742; T(2*n+1,n) = A016754(n);

T(n,0) = A000217(n);

T(n,1) = A000096(n) for n > 0;

T(n,2) = A055998(n) for n > 1;

T(n,3) = A055999(n) for n > 2;

T(n,4) = A056000(n) for n > 3;

T(n,5) = A056115(n) for n > 4;

T(n,6) = A056119(n) for n > 5;

T(n,7) = A056121(n) for n > 6;

T(n,8) = A056126(n) for n > 7;

T(n,10) = A101859(n-1) for n > 9;

T(n,n-3) = A095794(n-1) for n > 2;

T(n,n-2) = A045943(n-1) for n > 1;

T(n,n-1) = A000326(n) for n > 0;

T(n,n) = A005449(n).

Examples

			From _Philippe Deléham_, Oct 03 2011: (Start)
Triangle begins:
   0;
   1,  2;
   3,  5,  7;
   6,  9, 12, 15;
  10, 14, 18, 22, 26;
  15, 20, 25, 30, 35, 40;
  21, 27, 33, 39, 45, 51, 57;
  28, 35, 42, 49, 56, 63, 70, 77; (End)
		

References

• Léonard Euler, Introduction à l'analyse infinitésimale, tome premier, ACL-Editions, Paris, 1987, p. 353-354.

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• Émile Fourrey, Les nombres abstraits, Récreations arithmétiques, 1899 and later, Vuibert, Paris, page 86-87. Triangle without right diagonal.
• Adrien-Marie Legendre, Théorie des nombres, tome 2, quatrième partie, p.131, troisième édition, Paris, 1830.

Crossrefs

Cf. A110449.

Programs

Haskell

a126890 n k = a126890_tabl !! n !! k
a126890_row n = a126890_tabl !! n
a126890_tabl = map fst $ iterate
   (\(xs@(x:_), i) -> (zipWith (+) ((x-i):xs) [2*i+1 ..], i+1)) ([0], 0)
-- Reinhard Zumkeller, Nov 10 2013

Mathematica

Flatten[Table[(n(n+2k+1))/2,{n,0,20},{k,0,n}]] (* Harvey P. Dale, Jun 21 2013 *)

Formula

T(n,k) = T(n,k-1) + n, for k <= n. - Philippe Deléham, Oct 03 2011