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A115255 "Correlation triangle" of central binomial coefficients A000984.

%I A115255 #14 Aug 19 2025 08:43:39
%S A115255 1,2,2,6,5,6,20,14,14,20,70,46,41,46,70,252,160,134,134,160,252,924,
%T A115255 574,466,441,466,574,924,3432,2100,1672,1534,1534,1672,2100,3432,
%U A115255 12870,7788,6118,5506,5341,5506,6118,7788,12870,48620,29172,22692,20152,19174
%N A115255 "Correlation triangle" of central binomial coefficients A000984.
%C A115255 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 A115255 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
%F A115255 G.f.: 1/(sqrt(1-4*x)*sqrt(1-4*x*y)*(1-x^2*y)) (format due to _Christian G. Bower_).
%F A115255 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).
%e A115255 Triangle begins:
%e A115255   1;
%e A115255   2, 2;
%e A115255   6, 5, 6;
%e A115255   20, 14, 14, 20;
%e A115255   70, 46, 41, 46, 70;
%e A115255   252, 160, 134, 134, 160, 252;
%e A115255 Northwest corner (square format):
%e A115255   1    2    6    20    70
%e A115255   2    5    14   46    160
%e A115255   6    14   41   134   466
%e A115255   20   46   134  441   1534
%t A115255 s[k_] := Binomial[2 k - 2, k - 1];
%t A115255 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
%t A115255 L = Transpose[U]; M = L.U; TableForm[M]
%t A115255 m[i_, j_] := M[[i]][[j]]; (* A115255 in square format *)
%t A115255 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%t A115255 f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]; Table[f[n], {n, 1, 12}]
%t A115255 Table[Sqrt[f[n]], {n, 1, 12}]  (* A006134 *)
%t A115255 Table[m[1, j], {j, 1, 12}]     (* A000984 *)
%t A115255 Table[m[j, j], {j, 1, 12}]     (* A115257 *)
%t A115255 Table[m[j, j + 1], {j, 1, 12}] (* 2*A082578 *)
%t A115255 (* _Clark Kimberling_, Dec 27 2011 *)
%Y A115255 Cf. A203004, A203001, A202453.
%K A115255 easy,nonn,tabl
%O A115255 0,2
%A A115255 _Paul Barry_, Jan 18 2006