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A115255
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"Correlation triangle" of central binomial coefficients A000984.
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5
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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
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OFFSET
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0,2
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COMMENTS
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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]
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LINKS
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FORMULA
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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)}.
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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