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A144680 Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).

%I A144680 #7 Oct 20 2021 08:02:24
%S A144680 1,2,3,3,5,7,4,7,11,14,5,9,15,21,25,6,11,19,28,36,41,7,13,23,35,47,57,
%T A144680 63,8,15,27,42,58,73,85,92,9,17,31,49,69,89,107,121,129,10,19,35,56,
%U A144680 80,105,129,150,166,175
%N A144680 Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).
%C A144680 Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).
%H A144680 G. C. Greubel, <a href="/A144680/b144680.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A144680 Sum_{k=1..n} T(n, k) = A006008(n).
%F A144680 From _G. C. Greubel_, Oct 18 2021: (Start)
%F A144680 T(n, k) = (1/6)*( 3*(k^2 - k + 2)*n - k*(k-1)*(2*k-1) ).
%F A144680 T(n, n) = A004006(n).
%F A144680 T(n, n-1) = A050407(n+2).
%F A144680 T(n, n-2) = A027965(n-1) = A074742(n-2). (End)
%e A144680 The array is formed by A004736 * A144328 (transform) where A004736 = the natural number decrescendo triangle and A144328 = a crescendo triangle. First few rows of the array =
%e A144680   1, 1,  1,  1,  1,  1, ...
%e A144680   2, 3,  3,  3,  3,  3, ...
%e A144680   3, 5,  7,  7,  7,  7, ...
%e A144680   4, 7, 11, 14, 14, 14, ...
%e A144680   5, 9, 15, 21, 25, 25, ...
%e A144680   ...
%e A144680 Triangle begins as:
%e A144680    1;
%e A144680    2,  3;
%e A144680    3,  5,  7;
%e A144680    4,  7, 11, 14;
%e A144680    5,  9, 15, 21, 25;
%e A144680    6, 11, 19, 28, 36,  41;
%e A144680    7, 13, 23, 35, 47,  57,  63;
%e A144680    8, 15, 27, 42, 58,  73,  85,  92;
%e A144680    9, 17, 31, 49, 69,  89, 107, 121, 129;
%e A144680   10, 19, 35, 56, 80, 105, 129, 150, 166, 175;
%e A144680   ...
%t A144680 T[n_, k_]:= (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6;
%t A144680 Table[T[n, k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Oct 18 2021 *)
%o A144680 (Sage)
%o A144680 def A144680(n,k): return (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6
%o A144680 flatten([[A144680(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Oct 18 2021
%Y A144680 Cf. A004006, A006008, A027965, A050407, A074742, A144328.
%K A144680 nonn,tabl
%O A144680 1,2
%A A144680 _Gary W. Adamson_, Sep 19 2008