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A007971
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INVERTi transform of central trinomial coefficients (A002426).
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8
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0, 1, 2, 2, 4, 8, 18, 42, 102, 254, 646, 1670, 4376, 11596, 31022, 83670, 227268, 621144, 1706934, 4713558, 13072764, 36398568, 101704038, 285095118, 801526446, 2259520830, 6385455594, 18086805002, 51339636952, 146015545604
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OFFSET
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0,3
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COMMENTS
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For n>1, a(n) = 2(A005043(n-1)+A005043(n-2)). - Ralf Stephan, Jul 06 2003
Number of paths of a walk on the integers, allowing steps of size 0, +1, and -1, which return to the starting point for the first time at time n. [From David P. Sanders (dps(AT)fciencias.unam.mx), May 04 2009]
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LINKS
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Table of n, a(n) for n=0..29.
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FORMULA
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A002426(n) = Sum_{i=1..n} a(i)*A002426(n-i), n>0.
G.f.: 1-sqrt(1-2*x-3*x^2).
a(0)=0, a(1)=1, a(2)=2, then a(n)= (1/2) *(a(1)*a(n-1)+a(2)*a(n-2)+....+a(n-1)*a(1)) - Benoit Cloitre, Oct 24 2003
a(n)=2^(1-n)*sum(binomial(k,n-k)*a000108(k-1)*3^(n-k),k,1,n), n>0
[Vladimir Kruchinin, Feb 05 2011]
G.f.: 1-sqrt(1-2*x-3*(x^2))= x/G(0) ; G(k) = 1-2*x/(1+x/(1+x/(1-2*x/(1-x/(2-x/G(k+1)))))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011
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EXAMPLE
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x + 2*x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 18*x^6 + 42*x^7 + 102*x^8 + 254*x^9 + ...
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MATHEMATICA
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CoefficientList[Series[1-Sqrt[1-2x-3x^2], {x, 0, 40}], x] (* Harvey P. Dale, Dec 17 2012 *)
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CROSSREFS
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Cf. A002426. A001006(n)=A007971(n+2)/2.
Cf. A025227.
Sequence in context: * A126068 A167022 A168055 A005702 A095335 A100396
Adjacent sequences: A007968 A007969 A007970 * A007972 A007973 A007974
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KEYWORD
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nonn
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AUTHOR
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David Dumas (dumas(AT)TCNJ.EDU)
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EXTENSIONS
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Name corrected by Michael Somos, Mar 23 2012
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STATUS
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approved
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