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A119245
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Triangle, read by rows, defined by: T(n,k) = (4*k+1)*binomial(2*n+1, n-2*k)/(2*n+1) for n >= 2*k >= 0.
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4
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1, 1, 2, 1, 5, 5, 14, 20, 1, 42, 75, 9, 132, 275, 54, 1, 429, 1001, 273, 13, 1430, 3640, 1260, 104, 1, 4862, 13260, 5508, 663, 17, 16796, 48450, 23256, 3705, 170, 1, 58786, 177650, 95931, 19019, 1309, 21, 208012, 653752, 389367, 92092, 8602, 252, 1
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OFFSET
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0,3
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COMMENTS
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Closely related to triangle A118919.
Row n contains 1+floor(n/2) terms.
Combinatorial interpretations of T(n,k):
1) The number of standard tableaux of shape (n-2*k,n+2*k).
2) The entries in column k are (with an offset of 2*k) the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4*k. See [Sunik, Theorem 4]. (End)
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LINKS
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FORMULA
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G.f.: A(x,y) = f/(1-x^2*y*f^4), where f=(1-sqrt(1-4*x))/(2*x) is the Catalan g.f. (A000108).
Row sums equal A088218(n) = C(2*n-1,n).
T(n,0) = A000108(n) (the Catalan numbers).
Sum_{k=0..floor(n/2)} k*T(n,k) = A000346(n-2).
Eigenvector is defined by: A119244(n) = Sum_{k=0..[n\2]} T(n,k)*A119244(k).
...
T(n,k) = (4*k+1)/(n+2*k+1)*binomial(2*n,n+2*k). Compare with A158483. - Peter Bala, Mar 20 2009
A002894(n) = Sum_{k=0..floor(n/2)} (binomial(2k,k)^2)*(4^(n-2*k))*T(n,k). - Bradley Klee, Feb 26 2018
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EXAMPLE
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Triangle begins:
1;
1;
2, 1;
5, 5;
14, 20, 1;
42, 75, 9;
132, 275, 54, 1;
429, 1001, 273, 13;
1430, 3640, 1260, 104, 1;
4862, 13260, 5508, 663, 17; ...
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MATHEMATICA
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f1 = (1-Sqrt[1-4*x])/(2*x);
DeleteCases[CoefficientList[Normal@Series[f1/(1 - x^2*y*f1^4), {x, 0, 10}, {y, 0, 5}], {x, y}], 0, Infinity]//TableForm (* Bradley Klee, Feb 26 2018 *)
Table[(1+4*k)/(n+1+2*k)*Binomial[2*n, n+2*k], {n, 0, 10}, {k, 0, Floor[n/2]}]//TableForm (* Bradley Klee, Feb 26 2018 *)
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PROG
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(PARI) T(n, k)=(4*k+1)*binomial(2*n+1, n-2*k)/(2*n+1)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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