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Search: id:A014531
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| A014531 |
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Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 2nd column from the center. |
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11
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| 1, 3, 10, 30, 90, 266, 784, 2304, 6765, 19855, 58278, 171106, 502593, 1477035, 4343160, 12778152, 37616427, 110797569, 326527350, 962803170, 2840372304, 8383467708, 24755608584, 73133433800, 216143407675, 639062383401
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Number of "up" steps in all Motzkin paths of length n+1. E.g. a(2)=3 because in the four Motzkin paths of length 3, HHH, HUD, UDH, and UHD, where H=(1,0), U=(1,1), D=(1,-1), we have altogether three U steps. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2003
a(n-1) = A111808(n,n-2) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 17 2005
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..200
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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a(n) =A002426(n+1)-A001006(n+1) =a(n-1)+A005717(n)+A014532(n-2) - Henry Bottomley (se16(AT)btinternet.com), May 15 2001
E.g.f.: exp(x)*(2*x*BesselI(1, 2*x)+(x-2)*BesselI(2, 2*x))/x. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 21 2003
G.f.=[1-2z-z^2-(1-z)q]/(2z^3q), where q=sqrt(1-2z-3z^2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2003
a(n)=sum{k=0..n+1, binomial(n+1, k)binomial(n-k+1, k+2)} - Paul Barry (pbarry(AT)wit.ie), Sep 20 2004
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MAPLE
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seq( sum('binomial(i+1, k)*binomial(i-k+1, k+2)', 'k'=0..floor(i/2)), i=1..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
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CROSSREFS
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Cf. A027907.
First differences are in A025180.
Sequence in context: A094306 A026109 A026327 this_sequence A062107 A033113 A003441
Adjacent sequences: A014528 A014529 A014530 this_sequence A014532 A014533 A014534
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KEYWORD
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nonn,easy
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AUTHOR
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njas
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 05 2000
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