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A086615
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Antidiagonal sums of triangle A086614.
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9
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1, 2, 4, 8, 17, 38, 89, 216, 539, 1374, 3562, 9360, 24871, 66706, 180340, 490912, 1344379, 3701158, 10237540, 28436824, 79288843, 221836402, 622599625, 1752360040, 4945087837, 13988490338, 39658308814, 112666081616
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OFFSET
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0,2
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COMMENTS
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Partial sums of the Motzkin sequence (A001006). - Emeric Deutsch, Jul 12 2004
a(n) = number of distinct ordered trees obtained by branch-reducing the ordered trees on n+1 edges. - David Callan, Oct 24 2004
a(n)= the number of consecutive horizontal steps at height 0 of all Motzkin paths from (0,0) to (n,0) starting with a horizontal step. - Charles Moore (chamoore(AT)howard.edu), Apr 15 2007
Equals row sums of triangle A136788 - Gary W. Adamson, Jan 21 2008
The subsequence of prime partial sums of the Motzkin sequence begins: 2, 17, 89, no more through a(27). [From Jonathan Vos Post, Feb 11 2010]
This sequence (with offset 1 instead of 0) occurs in Section 7 of K. Grygiel, P. Lescanne (2015), see g.f. N. - N. J. A. Sloane, Nov 09 2015
Also number of plain (untyped) normal forms of lambda-terms (terms that cannot be further beta-reduced.) [Bendkowski et al., 2016]. - N. J. A. Sloane, Nov 22 2017
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
P. Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6
Maciej Bendkowski, K Grygiel, P Tarau, Random generation of closed simply-typed lambda-terms: a synergy between logic programming and Boltzmann samplers, arXiv preprint arXiv:1612.07682, 2016
K. Grygiel, P. Lescanne, A natural counting of lambda terms, SOFSEM 2016. Preprint 2015
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FORMULA
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G.f.: A(x) = 1/(1-x)^2 + x^2*A(x)^2.
a(n)=sum{k=0..floor((n+1)/2), binomial(n+1, 2k+1)binomial(2k, k)/(k+1)} - Paul Barry, Nov 29 2004
a(n) = n + 1 + sum_k a(k-1)a(n-k-1), starting from a(n)=0 for n negative. - Henry Bottomley, Feb 22 2005
a(n)=sum{k=0..n, sum{j=0..n-k, C(j)C(n-k, 2j)}}; - Paul Barry, Aug 19 2005
G.f.: c(x^2/(1-x)^2)/(1-x)^2, c(x) the g.f. of A000108; a(n)=sum{k=0..floor(n/2), C(n+1,n-2k)C(k)}; - Paul Barry, May 31 2006
Binomial transform of doubled Catalan sequence 1,1,1,1,2,2,5,5,14,14,... - Paul Barry, Nov 17 2005
Row sums of Pascal-Catalan triangle A086617. - Paul Barry, Nov 17 2005
g(z)=(1-z-sqrt(1-2z-3z^2))/(2z-2z^2)/z - Charles Moore (chamoore(AT)howard.edu), Apr 15 2007, corrected by Vaclav Kotesovec, Feb 13 2014
Conjecture: (n+2)*a(n) +3*(-n-1)*a(n-1) +(-n+4)*a(n-2) +3*(n-1)*a(n-3)=0. - R. J. Mathar, Nov 30 2012
a(n) ~ 3^(n+5/2) / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014
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EXAMPLE
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a(0)=1, a(1)=2, a(2)=3+1=4, a(3)=4+4=8, a(4)=5+10+2=17, a(5)=6+20+12=38, are upward antidiagonal sums of triangle A086614:
{1},
{2,1},
{3,4,2},
{4,10,12,5},
{5,20,42,40,14},
{6,35,112,180,140,42}, ...
For example with n=2, the 5 ordered trees (A000108) on 3 edges are
|...|..../\.../\.../|\..
|../.\..|......|........
|.......................
Suppressing nonroot vertices of outdegree 1 (branch-reducing) yields
|...|..../\.../\../|\..
.../.\.................
of which 4 are distinct. So a(2)=4.
a(4)=8 because we have HHHH, HHUD, HUDH, HUHD
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MATHEMATICA
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CoefficientList[Series[(1-x-Sqrt[1-2*x-3*x^2])/(2*x-2*x^2)/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)
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CROSSREFS
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Cf. A086614 (triangle), A086616 (row sums).
Cf. A001006.
Cf. A136788.
Sequence in context: A257300 A229202 A003007 * A081124 A090901 A101516
Adjacent sequences: A086612 A086613 A086614 * A086616 A086617 A086618
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jul 24 2003
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EXTENSIONS
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Edited by N. J. A. Sloane, Oct 16 2006
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STATUS
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approved
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