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A158822 Triangle read by rows, matrix triple product A000012 * A145677 * A000012.

%I A158822 #14 Dec 27 2021 08:24:33
%S A158822 1,3,1,6,3,2,10,6,5,3,15,10,9,7,4,21,15,14,12,9,5,28,21,20,18,15,11,6,
%T A158822 36,28,27,25,22,18,13,7,45,36,35,33,30,26,21,15,8,55,45,44,42,39,35,
%U A158822 30,24,17,9,66,55,54,52,49,45,40,34,27,19,10
%N A158822 Triangle read by rows, matrix triple product A000012 * A145677 * A000012.
%H A158822 G. C. Greubel, <a href="/A158822/b158822.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A158822 Triangle read by rows, A000012 * A145677 * A000012; where A000012 = an infinite lower triangular matrix: (1; 1,1; 1,1,1; ...), with all 1's.
%F A158822 From _G. C. Greubel_, Dec 26 2021: (Start)
%F A158822 T(n, k) = (n+1-k)*(n+k)/2 with T(n, 0) = binomial(n+2, 2).
%F A158822 Sum_{k=0..n} T(n, k) = (1/3)*(n+1)*(n^2 + 2*n + 3) = A006527(n+1).
%F A158822 Sum_{k=0..floor(n/2)} T(n-k, k) = binomial(n+2, 2) + A034828(n+1).
%F A158822 T(n, 1) = A000217(n).
%F A158822 T(n, 2) = A000096(n-1).
%F A158822 T(n, 3) = A055998(n-2).
%F A158822 T(2*n, n) = A134479(n). (End)
%e A158822 First few rows of the triangle =
%e A158822    1;
%e A158822    3,  1;
%e A158822    6,  3,  2;
%e A158822   10,  6,  5,  3;
%e A158822   15, 10,  9,  7,  4;
%e A158822   21, 15, 14, 12,  9,  5;
%e A158822   28, 21, 10, 18, 15, 11,  6;
%e A158822   36, 28, 27, 25, 22, 18, 13,  7;
%e A158822   45, 36, 35, 33, 30, 26, 21, 15,  8;
%e A158822   55, 45, 44, 42, 39, 35, 30, 24, 17,  9;
%e A158822   66, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10;
%e A158822   78, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21, 11;
%e A158822   91, 78, 77, 75, 72, 68, 63, 57, 50, 42, 33, 23, 12;
%e A158822   ...
%t A158822 T[n_, k_]:= If[k==0, Binomial[n+2, 2], (n+1-k)*(n+k)/2];
%t A158822 Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 26 2021 *)
%o A158822 (Sage)
%o A158822 def A158822(n,k):
%o A158822     if (k==0): return binomial(n+2, 2)
%o A158822     else: return (n-k+1)*(n+k)/2
%o A158822 flatten([[A158822(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Dec 26 2021
%Y A158822 Cf. A000012, A000096, A000217, A006527, A034828, A055998, A145677, A134479.
%K A158822 nonn,tabl
%O A158822 0,2
%A A158822 _Gary W. Adamson_ and _Roger L. Bagula_, Mar 28 2009
%E A158822 Definition corrected by _Michael Somos_, Nov 05 2011