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Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).
2

%I #7 Oct 20 2021 08:02:24

%S 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 63,8,15,27,42,58,73,85,92,9,17,31,49,69,89,107,121,129,10,19,35,56,

%U 80,105,129,150,166,175

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

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

%H G. C. Greubel, <a href="/A144680/b144680.txt">Rows n = 1..50 of the triangle, flattened</a>

%F Sum_{k=1..n} T(n, k) = A006008(n).

%F From _G. C. Greubel_, Oct 18 2021: (Start)

%F T(n, k) = (1/6)*( 3*(k^2 - k + 2)*n - k*(k-1)*(2*k-1) ).

%F T(n, n) = A004006(n).

%F T(n, n-1) = A050407(n+2).

%F T(n, n-2) = A027965(n-1) = A074742(n-2). (End)

%e 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 1, 1, 1, 1, 1, 1, ...

%e 2, 3, 3, 3, 3, 3, ...

%e 3, 5, 7, 7, 7, 7, ...

%e 4, 7, 11, 14, 14, 14, ...

%e 5, 9, 15, 21, 25, 25, ...

%e ...

%e Triangle begins as:

%e 1;

%e 2, 3;

%e 3, 5, 7;

%e 4, 7, 11, 14;

%e 5, 9, 15, 21, 25;

%e 6, 11, 19, 28, 36, 41;

%e 7, 13, 23, 35, 47, 57, 63;

%e 8, 15, 27, 42, 58, 73, 85, 92;

%e 9, 17, 31, 49, 69, 89, 107, 121, 129;

%e 10, 19, 35, 56, 80, 105, 129, 150, 166, 175;

%e ...

%t T[n_, k_]:= (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6;

%t Table[T[n, k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Oct 18 2021 *)

%o (Sage)

%o def A144680(n,k): return (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6

%o flatten([[A144680(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Oct 18 2021

%Y Cf. A004006, A006008, A027965, A050407, A074742, A144328.

%K nonn,tabl

%O 1,2

%A _Gary W. Adamson_, Sep 19 2008