cp's OEIS Frontend

A143156 Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.

%I A143156 #15 Oct 09 2021 07:22:47
%S A143156 1,3,2,4,3,1,7,6,4,3,8,7,5,4,1,10,9,7,6,3,2,11,10,8,7,4,3,1,15,14,12,
%T A143156 11,8,7,5,4,16,15,13,12,9,8,6,5,1,18,17,15,14,11,10,8,7,3,2,19,18,16,
%U A143156 15,12,11,9,8,4,3,1,22,21,19,18,15,14,12,11,7,6,4,3
%N A143156 Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.
%C A143156 Row sums give A143157.
%C A143156 Left border gives A005187.
%C A143156 Right border gives A001511.
%F A143156 Triangle read by rows, T(n,k) = Sum_{j=k..n} A001511(j); = A000012 * (A001511 * 0^(n-k)) * A000012; 1<=k<=n.
%F A143156 From _Kevin Ryde_, Oct 07 2021: (Start)
%F A143156 T(n,k) = A005187(n) - A005187(k-1).
%F A143156 G.f.: (V(x) - V(x*y)) * y/((1-x)*(1-y)) where V(x) is the g.f. of A001511.
%F A143156 (End)
%e A143156 First few rows of the triangle =
%e A143156         k=1 k=2 k=3 k=4 k=5 k=6 k=7
%e A143156   n=1:   1;
%e A143156   n=2:   3,  2;
%e A143156   n=3:   4,  3,  1;
%e A143156   n=4:   7,  6,  4,  3;
%e A143156   n=5:   8,  7,  5,  4,  1;
%e A143156   n=6:  10,  9,  7,  6,  3,  2;
%e A143156   n=7:  11, 10,  8,  7,  4,  3,  1;
%e A143156   ...
%e A143156 Row 6 = (10, 9, 7, 6, 3, 2) = partial sums of the first 6 terms of the ruler sequence, starting from the right: (1, 2, 1, 3, 1, 2,...).
%o A143156 (PARI) T(n,k) = k--; 2*(n-k) - hammingweight(n) + hammingweight(k); \\ _Kevin Ryde_, Oct 07 2021
%Y A143156 Cf. A001511, A000012, A091512, A005187.
%K A143156 nonn,easy,tabl
%O A143156 1,2
%A A143156 _Gary W. Adamson_, Jul 27 2008