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Search: id:A084773
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| A084773 |
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Coefficients of 1/sqrt(1-12*x+4*x^2); also, a(n) is the central coefficient of (1+6x+8x^2)^n. |
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+0
8
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| 1, 6, 52, 504, 5136, 53856, 575296, 6225792, 68026624, 748832256, 8291791872, 92255680512, 1030537089024, 11550176206848, 129824329777152, 1462841567576064, 16518691986407424, 186887008999047168
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where H and D can choose from two colors each (or where one step is monochrome and the other two are bicolored). - Paul Barry (pbarry(AT)wit.ie), May 30 2005
2^n*P_n(3), where P_n is the n-th Legendre polynomial. 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 (pbarry(AT)wit.ie), May 30 2005
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FORMULA
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E.g.f.: exp(6x)Bessel_I(0, 2*sqrt(8)x); a(n)=sum{k=0..floor(n/2), C(n, k)C(2(n-k), n)(-1)^k*3^(n-2k)}; - Paul Barry (pbarry(AT)wit.ie), May 30 2005
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EXAMPLE
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G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.
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PROGRAM
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(PARI) for(n=0, 30, t=polcoeff((1+6*x+8*x^2)^n, n, x); print1(t", "))
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CROSSREFS
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a(n) = 2*A052141(n) = 2^n * A001850(n), n>0.
See A152254 for another interpretation.
Sequence in context: A027111 A083301 A005948 this_sequence A127133 A075756 A144345
Adjacent sequences: A084770 A084771 A084772 this_sequence A084774 A084775 A084776
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jun 10 2003
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