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A026376
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a(n) = number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=2; also a(n)=T(2n,n-1).
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9
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1, 6, 30, 144, 685, 3258, 15533, 74280, 356283, 1713690, 8263596, 39938616, 193419915, 938430990, 4560542550, 22195961280, 108171753355, 527816696850, 2578310320610, 12607504827600, 61706212037295, 302275142049870
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OFFSET
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1,2
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COMMENTS
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Number of Schroeder paths (i.e. consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis) from (0,0) to (2n+2,0), with exactly one peak at an even level. E.g. a(2)=6 because we have UUDDH, HUUDD, UDUUDD, UUDDUD, UUDHD and UHUDD. - Emeric Deutsch, Dec 28 2003
Number of left steps in all skew Dyck paths of semilength n+1. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=6 because in the 10 (=A002212(3)) skew Dyck paths of semilength 3 ( namely UDUUDL, UUUDLD, UUDUDL, UUUDDL, UUUDLL and five Dyck paths that have no left steps) we have altogether 6 left steps. - Emeric Deutsch, Aug 05 2007
Contribution from Gary W. Adamson, May 17 2009: (Start)
Equals A026378: (1, 4, 17, 75, 339,...) convolved with A007317:
(1, 2, 5, 15, 51,...), and A081671: (1, 3, 11, 45, 195,...) convolved with
A002212: (1, 3, 10, 36, 137,...). (End)
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LINKS
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Table of n, a(n) for n=1..22.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
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FORMULA
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E.g.f.: exp(3x) I_1(2x), where I_1 is Bessel function. - Michael Somos, Sep 09 2002.
G.f.: [1-3z-sqrt(1-6z+5z^2)]/[2zsqrt(1-6z+5z^2)] - Emeric Deutsch, May 25 2003
a(n)=[t^(n+1)](1+3t+t^2)^n. a := n->sum(3^(2j-n-1)*binomial(n, j)*binomial(j, n+1-j), j=ceil((n+1)/2)..n). - Emeric Deutsch, Jan 30 2004
a(n)=sum{k=0..n, binomial(n, k)binomial(2k, k+1)} - Paul Barry, Sep 20 2004
a(n)=n*A002212(n). - Emeric Deutsch, Aug 05 2007
Conjecture: (n+1)*a(n) -9*n*a(n-1) +(23*n-27)*a(n-2) +15*(-n+2)*a(n-3)=0. - R. J. Mathar, Dec 02 2012
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PROG
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(PARI) a(n)=if(n<0, 0, polcoeff((1+3*x+x^2)^n, n-1))
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CROSSREFS
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Cf. A006318, A002212.
Cf. A081671, A007317, A026378 [From Gary W. Adamson, May 17 2009]
Sequence in context: A221397 A082134 A030192 * A026899 A135160 A046945
Adjacent sequences: A026373 A026374 A026375 * A026377 A026378 A026379
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling
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STATUS
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approved
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