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A119245
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Triangle, read by rows, defined by: T(n,k) = (4*k+1)*C(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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Closely related to triangle A118919. Row n contains 1+floor(n/2) terms. Row sums equal A088218(n) = C(2*n-1,n). T(n,0)=A000108(n) (the Catalan numbers). T(n,1)=A000344(n). T(n,2)=A001392(n). Sum_{k=0..[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).
Contribution from Peter Bala (pbala(AT)talktalk.net), Mar 20 2009: (Start)
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
| Zoran Sunik, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003). [From Peter Bala (pbala(AT)talktalk.net), Mar 20 2009]
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FORMULA
| G.f.: A(x,y) = C/(1-x*y^2*C^4), where C=[1-sqrt(1-4*x)]/(2*x) is the Catalan g.f. (A000108).
T(n,k) = (4*k+1)/(n+2*k+1)*binomial(2*n,n+2*k). Compare with A158483. [From Peter Bala (pbala(AT)talktalk.net), Mar 20 2009]
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EXAMPLE
| 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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PROG
| (PARI) T(n, k)=(4*k+1)*binomial(2*n+1, n-2*k)/(2*n+1)
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CROSSREFS
| Cf. A119244 (eigenvector), A088218, A000108, A000344, A001392; A118919 (variant).
Cf. A158483. [From Peter Bala (pbala(AT)talktalk.net), Mar 20 2009]
Sequence in context: A074392 A096976 A052547 * A128731 A129157 A086905
Adjacent sequences: A119242 A119243 A119244 * A119246 A119247 A119248
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), May 10 2006
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