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%I #8 Aug 03 2014 14:27:18
%S 1,2,2,6,5,6,20,14,14,20,70,46,41,46,70,252,160,134,134,160,252,924,
%T 574,466,441,466,574,924,3432,2100,1672,1534,1534,1672,2100,3432,
%U 12870,7788,6118,5506,5341,5506,6118,7788,12870,48620,29172,22692,20152,19174
%N "Correlation triangle" of central binomial coefficients A000984.
%C 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).
%C 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]
%F 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)}.
%e Triangle begins
%e 1;
%e 2, 2;
%e 6, 5, 6;
%e 20, 14, 14, 20;
%e 70, 46, 41, 46, 70;
%e 252, 160, 134, 134, 160, 252;
%e Northwest corner (square format):
%e 1....2....6....20....70
%e 2....5....14...46....160
%e 6....14...41...134...466
%e 20...46...134..441...1534
%t s[k_] := Binomial[2 k - 2, k - 1];
%t U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
%t L = Transpose[U]; M = L.U; TableForm[M]
%t m[i_, j_] := M[[i]][[j]]; (* A115255 in square format *)
%t Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%t f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]; Table[f[n], {n, 1, 12}]
%t Table[Sqrt[f[n]], {n, 1, 12}] (* A006134 *)
%t Table[m[1, j], {j, 1, 12}] (* A000984 *)
%t Table[m[j, j], {j, 1, 12}] (* A115257 *)
%t Table[m[j, j + 1], {j, 1, 12}] (* 2*A082578 *)
%t (* _Clark Kimberling_, Dec 27 2011 *)
%Y Cf. A203004, A203001, A202453.
%K easy,nonn,tabl
%O 0,2
%A _Paul Barry_, Jan 18 2006
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