A115255 "Correlation triangle" of central binomial coefficients A000984.

1, 2, 2, 6, 5, 6, 20, 14, 14, 20, 70, 46, 41, 46, 70, 252, 160, 134, 134, 160, 252, 924, 574, 466, 441, 466, 574, 924, 3432, 2100, 1672, 1534, 1534, 1672, 2100, 3432, 12870, 7788, 6118, 5506, 5341, 5506, 6118, 7788, 12870, 48620, 29172, 22692, 20152, 19174

Offset: 0

Author

Paul Barry, Jan 18 2006

Keywords

• easy
• nonn
• tabl
• changed

Comments

Row sums are A033114. Diagonal sums are A115256. T(2n,n) is A115257. Corresponds to the triangle of antidiagonals of the correlation matrix of the sequence array for C(2n,n).

Let s=(1,2,6,20,...), (central binomial coefficients), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A115255 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203005 for characteristic polynomials of principal submatrices of M, with interlacing zeros. - Clark Kimberling, Dec 27 2011

Examples

			Triangle begins:
  1;
  2, 2;
  6, 5, 6;
  20, 14, 14, 20;
  70, 46, 41, 46, 70;
  252, 160, 134, 134, 160, 252;
Northwest corner (square format):
  1    2    6    20    70
  2    5    14   46    160
  6    14   41   134   466
  20   46   134  441   1534
		

Crossrefs

Cf. A203004, A203001, A202453.

Programs

Mathematica

s[k_] := Binomial[2 k - 2, k - 1];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]]; (* A115255 in square format *)
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]; Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* A006134 *)
Table[m[1, j], {j, 1, 12}]     (* A000984 *)
Table[m[j, j], {j, 1, 12}]     (* A115257 *)
Table[m[j, j + 1], {j, 1, 12}] (* 2*A082578 *)
(* Clark Kimberling, Dec 27 2011 *)

Formula

G.f.: 1/(sqrt(1-4*x)*sqrt(1-4*x*y)*(1-x^2*y)) (format due to Christian G. Bower).

T(n, k) = Sum_{j=0..n} [j<=k]*C(2*k-2*j, k-j)*[j<=n-k]*C(2*n-2*k-2*j, n-k-j).