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A128177 A128174 * A004736 as infinite lower triangular matrices.

%I A128177 #21 Jan 05 2025 23:41:56
%S A128177 1,2,1,4,2,1,6,4,2,1,9,6,4,2,1,12,9,6,4,2,1,16,12,9,6,4,2,1,20,16,12,
%T A128177 9,6,4,2,1,25,20,16,12,9,6,4,2,1,30,25,20,16,12,9,6,4,2,1,36,30,25,20,
%U A128177 16,12,9,6,4,2,1,42,36,30,25,20,16,12,9,6,4,2,1
%N A128177 A128174 * A004736 as infinite lower triangular matrices.
%C A128177 n-th row has n nonzero terms of A002620: (1, 2, 4, 6, 9, 12, 16, ...) in reverse.
%C A128177 Row sums = A002623: (1, 3, 7, 13, 22, 34, 50, ...).
%F A128177 From _Ridouane Oudra_, Mar 23 2024: (Start)
%F A128177 T(n, k) = A002620(n-k+2), with 1 <= k <= n;
%F A128177 T(n, k) = floor((n-k+2)^2/4);
%F A128177 T(n, k) = (1/2)*floor((n-k+2)^2/2);
%F A128177 T(n, k) = (1/8)*(2*(n-k+2)^2 + (-1)^(n-k) - 1). (End)
%e A128177 First few rows of the triangle:
%e A128177    1;
%e A128177    2, 1;
%e A128177    4, 2, 1;
%e A128177    6, 4, 2, 1;
%e A128177    9, 6, 4, 2, 1;
%e A128177   12, 9, 6, 4, 2, 1;
%e A128177   ...
%p A128177 seq(seq(floor((n-k+2)^2/4), k=1..n), n=1..20); # _Ridouane Oudra_, Mar 23 2024
%t A128177 T[n_,k_]:=Floor[(n-k+2)^2/4];Table[T[n,k],{n,12},{k,n}]//Flatten (* _James C. McMahon_, Jan 05 2025 *)
%o A128177 (PARI) lista(nn) = {t128174 = matrix(nn, nn, n, k, (k<=n)*(1+(-1)^(n-k))/2); t004736 = matrix(nn, nn, n, k, (k<=n)*(n - k + 1)); t128177 = t128174*t004736; for (n = 1, nn, for (k = 1, n, print1(t128177[n, k], ", ");););} \\ _Michel Marcus_, Feb 11 2014
%Y A128177 Cf. A128174, A004736, A002623, A002620.
%K A128177 nonn,tabl
%O A128177 1,2
%A A128177 _Gary W. Adamson_, Feb 17 2007
%E A128177 Partially edited and more terms from _Michel Marcus_, Feb 11 2014