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

%I A126890 #38 Jan 22 2025 03:32:56
%S A126890 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 A126890 45,51,57,28,35,42,49,56,63,70,77,36,44,52,60,68,76,84,92,100,45,54,
%U A126890 63,72,81,90,99,108,117,126,55,65,75,85,95,105,115,125,135,145,155,66,77,88
%N A126890 Triangle read by rows: T(n,k) = n*(n+2*k+1)/2, 0 <= k <= n.
%C A126890 T(n,k) + T(n,n-k) = A014105(n);
%C A126890 row sums give A059270; Sum_{k=0..n-1} T(n,k) = A000578(n);
%C A126890 central terms give A007742; T(2*n+1,n) = A016754(n);
%C A126890 T(n,0)   = A000217(n);
%C A126890 T(n,1)   = A000096(n)   for n > 0;
%C A126890 T(n,2)   = A055998(n)   for n > 1;
%C A126890 T(n,3)   = A055999(n)   for n > 2;
%C A126890 T(n,4)   = A056000(n)   for n > 3;
%C A126890 T(n,5)   = A056115(n)   for n > 4;
%C A126890 T(n,6)   = A056119(n)   for n > 5;
%C A126890 T(n,7)   = A056121(n)   for n > 6;
%C A126890 T(n,8)   = A056126(n)   for n > 7;
%C A126890 T(n,10)  = A101859(n-1) for n > 9;
%C A126890 T(n,n-3) = A095794(n-1) for n > 2;
%C A126890 T(n,n-2) = A045943(n-1) for n > 1;
%C A126890 T(n,n-1) = A000326(n)   for n > 0;
%C A126890 T(n,n)   = A005449(n).
%D A126890 Léonard Euler, Introduction à l'analyse infinitésimale, tome premier, ACL-Editions, Paris, 1987, p. 353-354.
%H A126890 Reinhard Zumkeller, <a href="/A126890/b126890.txt">Rows n = 0..125 of triangle, flattened</a>
%H A126890 Émile Fourrey, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k875411n/f96.double">Les nombres abstraits</a>, Récreations arithmétiques, 1899 and later, Vuibert, Paris, page 86-87. Triangle without right diagonal.
%H A126890 Adrien-Marie Legendre, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k42612x/f146.item">Théorie des nombres</a>, tome 2, quatrième partie, p.131, troisième édition, Paris, 1830.
%F A126890 T(n,k) = T(n,k-1) + n, for k <= n. - _Philippe Deléham_, Oct 03 2011
%e A126890 From _Philippe Deléham_, Oct 03 2011: (Start)
%e A126890 Triangle begins:
%e A126890    0;
%e A126890    1,  2;
%e A126890    3,  5,  7;
%e A126890    6,  9, 12, 15;
%e A126890   10, 14, 18, 22, 26;
%e A126890   15, 20, 25, 30, 35, 40;
%e A126890   21, 27, 33, 39, 45, 51, 57;
%e A126890   28, 35, 42, 49, 56, 63, 70, 77; (End)
%t A126890 Flatten[Table[(n(n+2k+1))/2,{n,0,20},{k,0,n}]] (* _Harvey P. Dale_, Jun 21 2013 *)
%o A126890 (Haskell)
%o A126890 a126890 n k = a126890_tabl !! n !! k
%o A126890 a126890_row n = a126890_tabl !! n
%o A126890 a126890_tabl = map fst $ iterate
%o A126890    (\(xs@(x:_), i) -> (zipWith (+) ((x-i):xs) [2*i+1 ..], i+1)) ([0], 0)
%o A126890 -- _Reinhard Zumkeller_, Nov 10 2013
%Y A126890 Cf. A110449.
%K A126890 nonn,tabl,easy
%O A126890 0,3
%A A126890 _Reinhard Zumkeller_, Dec 30 2006