Revision History for A110616
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Showing entries 11-20
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#26 by Michael De Vlieger at Wed Jun 28 08:37:00 EDT 2017
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| MATHEMATICA
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Table[(k + 1) Binomial[3 n - 2 k, 2 n - k]/(2 n - k + 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jun 28 2017 *)
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proposed
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#25 by Michel Marcus at Wed Jun 28 08:04:19 EDT 2017
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#24 by Michel Marcus at Wed Jun 28 08:04:05 EDT 2017
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| COMMENTS
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With offset 1 for n and k, T(n,k) = number of Dyck paths of semilength n for which all descents are of even length (counted by A001764) with no valley vertices at height 1 and with k returns to ground level. For example, T(3,2)=2 counts U^4 D^4 U^2 D^2, U^2 D^2 U^4 D^4 where U=upstep, D=downstep and exponents denote repetition. [From _. [_David Callan_, Aug 27 2009]
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| REFERENCES
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Naiomi Cameron, JE McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
Sheng-Liang Yang, LJ Wang, Taylor expansions for the m-Catalan numbers, AUSTRALASIAN JOURNAL OF COMBINATORICS, Volume 64(3) (2016), Pages 420-431.
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| LINKS
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Naiomi Cameron, J. E. McLeod, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/McLeod/mcleod3.html">Returns and Hills on Generalized Dyck Paths</a>, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
Sheng-Liang Yang, LJ Wang, <a href="https://ajc.maths.uq.edu.au/pdf/64/ajc_v64_p420.pdf">Taylor expansions for the m-Catalan numbers</a>, Australasian Journal of Combinatorics, Volume 64(3) (2016), Pages 420-431.
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| FORMULA
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GF: 1/(1 - x*y*TernaryGF) = 1 + (y)x + (y+y^2)x^2 + (3y+2y^2+y^3)x^3 +... where TernaryGF = 1 + x + 3x^2 + 12x^3 +... is the GF for A001764. [From _. [_David Callan_, Aug 27 2009]
T(n, k) = ((k+1)*binomial(3*n-2*k,2*n-k))/(2*n-k+1); [From ). [_Vladimir Kruchinin, _, Nov 01 2011]
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246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1 ;...;
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| PROG
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(Maxima)) T(n, k):=((k+1)*binomial(3*n-2*k, 2*n-k))/(2*n-k+1); // _Vladimir Kruchinin_, Nov 01 2011
T(n, k):=((k+1)*binomial(3*n-2*k, 2*n-k))/(2*n-k+1); [From Vladimir Kruchinin, Nov 01 2011]
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approved
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#23 by N. J. A. Sloane at Sun May 21 01:10:15 EDT 2017
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#22 by N. J. A. Sloane at Sun May 21 01:10:11 EDT 2017
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| REFERENCES
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Naiomi Cameron, JE McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
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approved
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#21 by N. J. A. Sloane at Mon Jan 16 18:47:13 EST 2017
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#20 by N. J. A. Sloane at Mon Jan 16 18:47:11 EST 2017
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| REFERENCES
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Sheng-Liang Yang, LJ Wang, Taylor expansions for the m-Catalan numbers, AUSTRALASIAN JOURNAL OF COMBINATORICS, Volume 64(3) (2016), Pages 420-431.
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approved
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#19 by Joerg Arndt at Mon Jan 27 04:41:18 EST 2014
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#18 by Philippe Deléham at Sun Jan 26 22:22:17 EST 2014
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#17 by Philippe Deléham at Sun Jan 26 22:22:08 EST 2014
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| COMMENTS
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Riordan array (f(x), x*f(x)) with f(x) = (2/sqrt(3*x))*sin((1/3)*arcsin(sqrt(27*x/4))). - Philippe Deléham, Jan 27 2014
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approved
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