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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 February 17 17:59 EST 2020. Contains 331999 sequences. (Running on oeis4.)