cp's OEIS Frontend

A115262 Correlation triangle for n+1.

%I A115262 #19 Jan 21 2020 21:00:22
%S A115262 1,2,2,3,5,3,4,8,8,4,5,11,14,11,5,6,14,20,20,14,6,7,17,26,30,26,17,7,
%T A115262 8,20,32,40,40,32,20,8,9,23,38,50,55,50,38,23,9,10,26,44,60,70,70,60,
%U A115262 44,26,10,11,29,50,70,85,91,85,70,50,29,11
%N A115262 Correlation triangle for n+1.
%C A115262 This sequence (formatted as a square array) gives the counts of all possible squares in an m X n rectangle. For example, 11 = 8 (1 X 1 squares) + 3 (2 X 2 square) in 4 X 2 rectangle. - _Philippe Deléham_, Nov 26 2009
%C A115262 From _Clark Kimberling_, Feb 07 2011: (Start)
%C A115262 Also the accumulation array of min{n,k}, when formatted as a rectangle.
%C A115262 This is the accumulation array of the array M=A003783 given by M(n,k)=min{n,k}; see A144112 for the definition of accumulation array.
%C A115262 The accumulation array of A115262 is A185957. (End)
%C A115262 From _Clark Kimberling_, Dec 22 2011: (Start)
%C A115262 As a square matrix, A115262 is the self-fusion matrix of A000027 (1,2,3,4,...).  See A193722 for the definition of fusion and A202673 for characteristic polynomials associated with A115622. (End)
%F A115262 Let f(m,n) = m*(m-1)*(3*n-m-1)/6. This array is (with a different offset) the infinite square array read by antidiagonals U(m,n) = f(n,m) if m < n, U(m,n) = f(m,n) if m <= n. See A271916. - _N. J. A. Sloane_, Apr 26 2016
%F A115262 G.f.: 1/((1-x)^2*(1-x*y)^2*(1-x^2*y)).
%F A115262 Number triangle T(n, k) = Sum_{j=0..n} [j<=k]*(k-j+1)[j<=n-k]*(n-k-j+1).
%F A115262 T(2n,n) - T(2n,n+1) = n+1.
%e A115262 Triangle begins
%e A115262   1;
%e A115262   2,  2;
%e A115262   3,  5,  3;
%e A115262   4,  8,  8,  4;
%e A115262   5, 11, 14, 11,  5;
%e A115262   6, 14, 20, 20, 14,  6;
%e A115262   ...
%e A115262 When formatted as a square matrix:
%e A115262   1,  2,  3,  4,  5, ...
%e A115262   2,  5,  8, 11, 14, ...
%e A115262   3,  8, 14, 20, 26, ...
%e A115262   4, 11, 20, 30, 40, ...
%e A115262   5, 14, 26, 40, 55, ...
%e A115262   ...
%t A115262 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k, {k, 1, 12}]];
%t A115262 L = Transpose[U]; M = L.U; TableForm[M]
%t A115262 m[i_, j_] := M[[i]][[j]];
%t A115262 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%t A115262 (* _Clark Kimberling_, Dec 22 2011 *)
%Y A115262 Cf. A000027, A202673, A271916.
%Y A115262 For the triangular version: row sums are A001752. Diagonal sums are A097701. T(2n,n) is A000330(n+1).
%Y A115262 Diagonals (1,5,...): A000330 (square pyramidal numbers),
%Y A115262 diagonals (2,8,...): A007290,
%Y A115262 diagonals (3,11,...): A051925,
%Y A115262 diagonals (4,14,...): A159920,
%Y A115262 antidiagonal sums: A001752.
%K A115262 easy,nonn,tabl
%O A115262 0,2
%A A115262 _Paul Barry_, Jan 18 2006