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A069835
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Define an array as follows: b(i,0)=b(0,j)=1, b(i,j)=2*b(i-1,j-1)+b(i-1,j)+b(i,j-1). Then a(n)=b(n,n).
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10
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1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544, 98365972, 700701808, 5011371964, 35961808432, 258805997752, 1867175631136, 13500088649734, 97794850668952, 709626281415076, 5157024231645616, 37528209137458516, 273431636191026064, 1994448720786816712
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OFFSET
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0,2
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COMMENTS
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2^n*LegendreP(n,k) yields the central coefficients of (1+2kx+(k^2-1)x^2)^n, with g.f. 1/sqrt(1-4kx+4x^2) and e.g.f. exp(2kx)BesselI(0,2sqrt(k^2-1)x). - Paul Barry, May 25 2005
Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where D can have two colors. - Paul Barry, May 25 2005
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps can have three colors and H steps can have four colors. - N-E. Fahssi, Mar 31 2008
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and two kinds of steps (1,1). [Joerg Arndt, Jul 01 2011]
Hankel transform is 2^n*3^C(n+1,2)=(-1)^C(n+1,2)*A127946(n). [From Paul Barry, 24 Jan 2011]
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REFERENCES
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Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.- From N. J. A. Sloane, Oct 08 2012
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LINKS
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Arkadiusz Wesolowski, Table of n, a(n) for n = 0..250
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
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FORMULA
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a(n) = 2^n*LegendreP(n, 2) = 2^n*hypergeom([ -n, n+1], [1], -1/2) = 2^n*GegenbauerC(n, 1/2, 2) = Sum_{k=0..n} 3^k*binomial(n, k)^2. Recurrence: a(n) = 4*(2*n-1)/n*a(n-1)-4*(n-1)/n*a(n-2). G.f.: 1/sqrt(1-8*x+4*x^2). - Vladeta Jovovic, May 13 2003
a(n) = central coefficient of (1+4*x+3*x^2)^n. - Paul D. Hanna, Jun 03 2003
E.g.f.: exp(4*x)*Bessel_I(0, 2*sqrt(3)*x) - Paul Barry, Sep 20 2004
a(n) = sum(k=0..floor(n/2), C(n, k)*C(2*(n-k), n)*(-1)^k*2^(n-2*k) ); - Paul Barry, May 25 2005
a(n) = sum(k=0..n, C(n, k)*C(n+k, k)*2^(n-k) ). - Paul Barry, May 25 2005
a(n) = sum(k=0..n, C(n, k)^2*3^k ). - Paul Barry, Oct 15 2005
G.f.: 1/(1-4x-6x^2/(1-4x-3x^2/(1-4x-3x^2/(1-4x-3x^2/(1-... (continued fraction). [From Paul Barry Jan 24 2011]
Asymptotic: a(n) ~ sqrt(1/2+1/sqrt(3))*(1+sqrt(3))^(2*n)/sqrt(Pi*n). [Vaclav Kotesovec, Sep 11 2012]
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EXAMPLE
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The array b is a rewriting of A081577:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,..
1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31,...
1, 7, 22, 46, 79,121,172,232,301,379,466,...
1, 10, 46,136,307,586,1000,1576,2341,3322,4546,...
1, 13, 79,307,886,2086,4258,7834,13327,21331,32521,...
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MATHEMATICA
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Table[Hypergeometric2F1[-n, -n, 1, 3], {n, 0, 21}] (* Arkadiusz Wesolowski, Aug 13 2012 *)
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PROG
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(PARI) a(n)=sum(k=0, n, binomial(n, k)^2*3^k)
(PARI) {a(n)=if(n<0, 0, polcoeff((1+4*x+3*x^2)^n, n))}
(PARI) /* as lattice paths: same as in A092566 but use */
steps=[[1, 0], [0, 1], [1, 1], [1, 1]]; /* note the double [1, 1] */
/* Joerg Arndt, Jul 01 2011 */
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CROSSREFS
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Cf. A001850.
Sequence in context: A183281 A067120 A143648 * A007196 A091638 A142984
Adjacent sequences: A069832 A069833 A069834 * A069836 A069837 A069838
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KEYWORD
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easy,nonn,changed
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AUTHOR
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Benoit Cloitre, May 03 2002
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STATUS
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approved
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