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 A126890 Triangle read by rows: T(n,k) = n*(n+2*k+1)/2, 0 <= k <= n. 20
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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). REFERENCES Léonard Euler, Introduction à l'analyse infinitésimale, tome premier, ACL-Editions, Paris, 1987, p. 353-354 Emile Fourrey, Récréations arithmétiques, Librairie Vuibert, Paris 1899, p. 96 Adrien-Marie Legendre, Théorie des nombres, tome 2, quatrième partie, p.131, troisième édition, Paris, 1830. LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened FORMULA T(n,k) = T(n,k-1) + n, for k <= n. - Philippe Deléham, Oct 03 2011 EXAMPLE 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) MATHEMATICA Flatten[Table[(n(n+2k+1))/2, {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, Jun 21 2013 *) PROG (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 CROSSREFS Cf. A110449. Sequence in context: A064143 A283593 A115274 * A122637 A076229 A160102 Adjacent sequences: A126887 A126888 A126889 * A126891 A126892 A126893 KEYWORD nonn,tabl,easy AUTHOR Reinhard Zumkeller, Dec 30 2006 STATUS approved

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Last modified December 3 18:25 EST 2022. Contains 358539 sequences. (Running on oeis4.)