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A128623 Triangle read by rows: A128621 * A000012 as infinite lower triangular matrices.

%I A128623 #19 Mar 15 2024 12:49:50
%S A128623 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 A128623 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 A128623 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 A128623 Triangle read by rows: A128621 * A000012 as infinite lower triangular matrices.
%H A128623 G. C. Greubel, <a href="/A128623/b128623.txt">Rows n = 1..100 of the triangle, flattened</a>
%F A128623 Sum_{k=1..n} T(n, k) = A128624(n) (row sums).
%F A128623 T(n,k) = n*(1+floor((n-k)/2)), 1 <= k <= n. - _R. J. Mathar_, Jun 27 2012
%F A128623 From _G. C. Greubel_, Mar 13 2024: (Start)
%F A128623 T(n, k) = n*A115514(n, k).
%F A128623 T(n, k) = Sum_{j=k..n} A128621(n, j).
%F A128623 T(n, 1) = A093005(n).
%F A128623 T(n, 2) = A093353(n-1), n >= 2.
%F A128623 T(n, n) = A000027(n).
%F A128623 T(2*n-1, n) = A245524(n).
%F A128623 Sum_{k=1..n} (-1)^k*T(n, k) = (1/2)*(1-(-1)^n)*A000384(floor((n+1)/2)). (End)
%e A128623 First few rows of the triangle are:
%e A128623    1;
%e A128623    2,  2;
%e A128623    6,  3,  3;
%e A128623    8,  8,  4,  4;
%e A128623   15, 10, 10,  5,  5;
%e A128623   18, 18, 12, 12,  6, 6;
%e A128623   28, 21, 21, 14, 14, 7, 7;
%e A128623   ...
%t A128623 Table[n*Floor[(n-k+2)/2], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 13 2024 *)
%o A128623 (Magma) [n*Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Mar 13 2024
%o A128623 (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 A128623 Cf. A000027, A093005, A093353, A115514, A128621, A245524.
%Y A128623 Cf. A128624 (row sums).
%K A128623 nonn,easy,tabl
%O A128623 1,2
%A A128623 _Gary W. Adamson_, Mar 14 2007
%E A128623 a(41) = 27 inserted and more terms from _Georg Fischer_, Jun 05 2023