%I #19 Jan 21 2020 21:00:22

%S 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 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 44,26,10,11,29,50,70,85,91,85,70,50,29,11

%N Correlation triangle for n+1.

%C 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 From _Clark Kimberling_, Feb 07 2011: (Start)

%C Also the accumulation array of min{n,k}, when formatted as a rectangle.

%C 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 The accumulation array of A115262 is A185957. (End)

%C From _Clark Kimberling_, Dec 22 2011: (Start)

%C 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 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 G.f.: 1/((1-x)^2*(1-x*y)^2*(1-x^2*y)).

%F Number triangle T(n, k) = Sum_{j=0..n} [j<=k]*(k-j+1)[j<=n-k]*(n-k-j+1).

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

%e Triangle begins

%e 1;

%e 2, 2;

%e 3, 5, 3;

%e 4, 8, 8, 4;

%e 5, 11, 14, 11, 5;

%e 6, 14, 20, 20, 14, 6;

%e ...

%e When formatted as a square matrix:

%e 1, 2, 3, 4, 5, ...

%e 2, 5, 8, 11, 14, ...

%e 3, 8, 14, 20, 26, ...

%e 4, 11, 20, 30, 40, ...

%e 5, 14, 26, 40, 55, ...

%e ...

%t U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k, {k, 1, 12}]];

%t L = Transpose[U]; M = L.U; TableForm[M]

%t m[i_, j_] := M[[i]][[j]];

%t Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

%t (* _Clark Kimberling_, Dec 22 2011 *)

%Y Cf. A000027, A202673, A271916.

%Y For the triangular version: row sums are A001752. Diagonal sums are A097701. T(2n,n) is A000330(n+1).

%Y Diagonals (1,5,...): A000330 (square pyramidal numbers),

%Y diagonals (2,8,...): A007290,

%Y diagonals (3,11,...): A051925,

%Y diagonals (4,14,...): A159920,

%Y antidiagonal sums: A001752.

%K easy,nonn,tabl

%O 0,2

%A _Paul Barry_, Jan 18 2006