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

0,2

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. [From Clark Kimberling, Dec 27 2011]

Links

Table of n, a(n) for n=0..49.

Formula

G.f.: 1/(sqrt(1-4x)*sqrt(1-4x*y)*(1-x^2*y)) (format due to Christian G. Bower); Number triangle T(n, k)=sum{j=0..n, [j<=k]*C(2k-2j, k-j)[j<=n-k]*C(2n-2k-2j, n-k-j)}.

Example

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

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 *)

Crossrefs

Cf. A203004, A203001, A202453.

Sequence in context: A200226 A300628 A379822 * A055924 A286278 A156563

Adjacent sequences: A115252 A115253 A115254 * A115256 A115257 A115258

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 18 2006

Status

approved