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 A126890 Triangle read by rows: T(n,k) = n*(n+2*k+1)/2, 0<=k<=n. 20

%I

%S 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,

%T 45,51,57,28,35,42,49,56,63,70,77,36,44,52,60,68,76,84,92,100,45,54,

%U 63,72,81,90,99,108,117,126,55,65,75,85,95,105,115,125,135,145,155,66,77,88

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

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

%C row sums give A059270; Sum(T(n,k): 0<=k<n) = A000578(n);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%D Emile Fourrey, Récréations arithmétiques, Librairie Vuibert, Paris 1899, p. 96

%D Adrien-Marie Legendre, Théorie des nombres, tome 2, quatrième partie, p.131, troisième édition, Paris, 1830.

%H Reinhard Zumkeller, <a href="/A126890/b126890.txt">Rows n = 0..125 of triangle, flattened</a>

%F T(n,k)=T(n,k-1)+n ,for k<=n [from _Philippe Deléham_, Oct 03 2011]

%e Triangle begins :

%e 0 ;

%e 1, 2 ;

%e 3, 5, 7 ;

%e 6, 9, 12, 15 ;

%e 10, 14, 18, 22, 26 ;

%e 15, 20, 25, 30, 35, 40 ;

%e 21, 27, 33, 39, 45, 51, 57 ;

%e 28, 35, 42, 49, 56, 63, 70, 77 ;... [from _Philippe Deléham_, Oct 03 2011]

%t Flatten[Table[(n(n+2k+1))/2,{n,0,20},{k,0,n}]] (* _Harvey P. Dale_, Jun 21 2013 *)

%o a126890 n k = a126890_tabl !! n !! k

%o a126890_row n = a126890_tabl !! n

%o a126890_tabl = map fst \$ iterate

%o (\(xs@(x:_), i) -> (zipWith (+) ((x-i):xs) [2*i+1 ..], i+1)) ([0], 0)

%o -- _Reinhard Zumkeller_, Nov 10 2013

%Y Cf. A110449.

%K nonn,tabl,easy

%O 0,3

%A _Reinhard Zumkeller_, Dec 30 2006

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Last modified February 23 05:58 EST 2019. Contains 320411 sequences. (Running on oeis4.)