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Triangle read by rows: A128621 * A000012 as infinite lower triangular matrices.
3

%I #19 Mar 15 2024 12:49:50

%S 1,2,2,6,3,3,8,8,4,4,15,10,10,5,5,18,18,12,12,6,6,28,21,21,14,14,7,7,

%T 32,32,24,24,16,16,8,8,45,36,36,27,27,18,18,9,9,50,50,40,40,30,30,20,

%U 20,10,10,66,55,55,44,44,33,33,22,22,11,11,72,72,60,60,48,48,36,36,24,24,12,12,91,78,78,65,65,52,52,39,39,26,26,13,13

%N Triangle read by rows: A128621 * A000012 as infinite lower triangular matrices.

%H G. C. Greubel, <a href="/A128623/b128623.txt">Rows n = 1..100 of the triangle, flattened</a>

%F Sum_{k=1..n} T(n, k) = A128624(n) (row sums).

%F T(n,k) = n*(1+floor((n-k)/2)), 1 <= k <= n. - _R. J. Mathar_, Jun 27 2012

%F From _G. C. Greubel_, Mar 13 2024: (Start)

%F T(n, k) = n*A115514(n, k).

%F T(n, k) = Sum_{j=k..n} A128621(n, j).

%F T(n, 1) = A093005(n).

%F T(n, 2) = A093353(n-1), n >= 2.

%F T(n, n) = A000027(n).

%F T(2*n-1, n) = A245524(n).

%F Sum_{k=1..n} (-1)^k*T(n, k) = (1/2)*(1-(-1)^n)*A000384(floor((n+1)/2)). (End)

%e First few rows of the triangle are:

%e 1;

%e 2, 2;

%e 6, 3, 3;

%e 8, 8, 4, 4;

%e 15, 10, 10, 5, 5;

%e 18, 18, 12, 12, 6, 6;

%e 28, 21, 21, 14, 14, 7, 7;

%e ...

%t Table[n*Floor[(n-k+2)/2], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 13 2024 *)

%o (Magma) [n*Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Mar 13 2024

%o (SageMath) flatten([[n*((n-k+2)//2) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Mar 13 2024

%Y Cf. A000027, A093005, A093353, A115514, A128621, A245524.

%Y Cf. A128624 (row sums).

%K nonn,easy,tabl

%O 1,2

%A _Gary W. Adamson_, Mar 14 2007

%E a(41) = 27 inserted and more terms from _Georg Fischer_, Jun 05 2023