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Search: id:A005573
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| A005573 |
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Number of walks on cubic lattice (starting from origin and not going below xy plane).
(Formerly M3943)
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+0
6
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| 1, 5, 26, 139, 758, 4194, 23460, 132339, 751526, 4290838, 24607628, 141648830, 817952188, 4736107172, 27487711752, 159864676803, 931448227590, 5435879858958, 31769632683132, 185918669183370, 1089302293140564
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Binomial transform of A026378, second binomial transform of A001700 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 28 2007
The Hankel transform of [1,1,5,26,139,758,...] is [1,4,15,56,209,...](see A001353). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 13 2007
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
E. Deutsch et al., Problem 10795, Amer. Math. Monthly, 108 (Dec. 2001), 980.
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LINKS
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R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
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FORMULA
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sum((-1)^i*6^(n-i)*binomial(n, i)*binomial(2*i, i)/(i+1), i=0..n); g.f. A(x) satisfies x(1-6x)A^2+(1-6x)A-1=0 - from Emeric Deutsch (deutsch(AT)duke.poly.edu); corrected by Roland Bacher (Roland.Bacher(AT)ujf-grenoble.fr), Jan 09 2003
a(n) = 6a(n-1)-A005572(n-1) = sum{j = 0, ..., n}[4^(n-j)*C(n, [n/2])*C(n, j)] - Henry Bottomley (se16(AT)btinternet.com), Aug 23 2001
a(n) = sum_{k=0..n} binomial(n, k)*binomial(2*k+1, k)*2^(n-k).
a(n) = sum_{k=0..n} (-1)^k*binomial(n, k)*Catalan(k)*6^(n-k).
(n+1)*a(n) = (8*n+2)*a(n-1)-(12*n-12)*a(n-2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 16 2004
a(n) = Sum_{k, 0<=k<=n} A052179(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 28 2007
Contribution from Paul Barry (pbarry(AT)wit.ie), Apr 21 2009: (Start)
G.f.: (sqrt((1-2x)/(1-6x))-1)/(2x);
G.f.: 1/(1-5x-x^2/(1-4x-x^2/(1-4x-x^2/(1-4x-x^2/(1-... (continued fraction). (End)
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CROSSREFS
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Sequence in context: A049607 A035029 A081569 this_sequence A081911 A081187 A104498
Adjacent sequences: A005570 A005571 A005572 this_sequence A005574 A005575 A005576
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Henry Bottomley (se16(AT)btinternet.com), Aug 23 2001
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