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A137644
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a(n) = Sum_{k=0..n} C(n+k,k)*C(n+k,n-k).
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8
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1, 3, 16, 95, 591, 3780, 24620, 162423, 1081780, 7258053, 48982176, 332140328, 2261099491, 15444137880, 105789736896, 726426836103, 4998885106599, 34464824536500, 238017084356680, 1646234203000485, 11401464090042224
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OFFSET
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0,2
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COMMENTS
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Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (0,1), (0,2). [From Eric Werley, Jun 29 2011]
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LINKS
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Table of n, a(n) for n=0..20.
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FORMULA
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a(n)= 3F2( {-n, n+1, n+1}; {1/2, 1})( -(1/4) ) [From Olivier GERARD, Apr 23 2009]
G.f. A(x)=F'(x)/(1+F(x)), F(x)=x*(1+F(x))/(1-F(x)-F(x)^2). [From Vladimir Kruchinin, Mar 24 2012]
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EXAMPLE
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The triangle of number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (0,1), (0,2) begins:
1;
1, 3;
1, 5, 16;
1, 7, 29, 95;
1, 9, 46, 179, 591;
1, 11, 67, 303, 1140, 3780;
1, 13, 92, 475, 2010, 7405, 24620;
1, 15, 121, 703, 3309, 13427, 48761, 162423;
1, 17, 154, 995, 5161, 22892, 90241, 324317, 1081780;
This sequence is the diagonal. [Joerg Arndt, Jul 01, 2011]
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MATHEMATICA
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Table[ HypergeometricPFQ[{-n, 1 + n, 1 + n}, {1/2, 1}, -(1/4)], {n, 0, 20}] [From Olivier GERARD, Apr 23 2009]
Table[Sum[Binomial[n+k, k]Binomial[n+k, n-k], {k, 0, n}], {n, 0, 20}] (* From Harvey P. Dale, Aug 03 2011 *)
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PROG
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(PARI) a(n)=sum(k=0, n, binomial(n+k, k)*binomial(n+k, n-k))
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CROSSREFS
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(PARI) /* same as in A092566 but use */
steps=[[1,0], [1,1], [0,1], [0,2]];
/* Joerg Arndt, Jun 30 2011 */
Sequence in context: A221764 A213229 A074555 * A114174 A181067 A006347
Adjacent sequences: A137641 A137642 A137643 * A137645 A137646 A137647
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jan 31 2008
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STATUS
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approved
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