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a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*2^i*3^(n-i), a(0)=1.
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%I #99 Aug 16 2024 06:01:48

%S 1,3,15,93,645,4791,37275,299865,2474025,20819307,178003815,

%T 1541918901,13503125805,119352115551,1063366539315,9539785668657,

%U 86104685123025,781343125570515,7124072211203775,65233526296899981,599633539433039445,5531156299278726663

%N a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*2^i*3^(n-i), a(0)=1.

%C The Hankel transform of this sequence is 6^C(n+1,2). - _Philippe Deléham_, Oct 28 2007

%C The Hankel transform of the sequence starting 1, 1, 3, 15, ... is A081955. - _Paul Barry_, Dec 09 2008

%C Number of Schroeder paths from (0,0) to (0,2n) allowing two colors for the down steps (or alternatively for the rise steps). - _Paul Barry_, Feb 01 2009

%C Essentially, reversion of x*(1-2*x)/(1+x). - _Paul Barry_, Apr 28 2009

%C a(n) is also the number of infix expressions with n variables and operators +, - and * (or +, * and /) such that there are no redundant parentheses. - _Vjeran Crnjak_, Apr 25 2020

%H Vincenzo Librandi, <a href="/A103210/b103210.txt">Table of n, a(n) for n = 0..200</a>

%H Joseph Abate and Ward Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Whitt/whitt2.html">Integer Sequences from Queueing Theory </a>, J. Int. Seq. 13 (2010), 10.5.5. b_n(2).

%H Eyal Ackerman, Gill Barequet, Ron Y. Pinter, and Dan Romik, <a href="https://doi.org/10.1016/j.ipl.2006.01.011">The number of guillotine partitions in d dimensions</a>, Inf. Proc. Lett. (2006) Vol. 98, No. 4, 162-167.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.

%H Zhi Chen and Hao Pan, <a href="http://arxiv.org/abs/1608.02448">Identities involving weighted Catalan-Schroder and Motzkin Paths</a>, arXiv:1608.02448 [math.CO], 2016, eq. (1.13), a=3, b=2.

%H Robert Dickau, <a href="https://robertdickau.com/schroederslices3d.html">3D Guillotine Partitions</a>, figures for 3D slices.

%H Samuele Giraudo, <a href="http://arxiv.org/abs/1504.04529">Operads from posets and Koszul duality</a>, arXiv preprint arXiv:1504.04529 [math.CO], 2015.

%H Samuele Giraudo, <a href="http://arxiv.org/abs/1603.01394">Pluriassociative algebras II: The polydendriform operad and related operads</a>, arXiv:1603.01394 [math.CO], 2016.

%H Luis Verde-Star, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Verde/verde4.html">A Matrix Approach to Generalized Delannoy and Schröder Arrays</a>, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

%F G.f.: (1 - z - sqrt(1 -10*z +z^2))/(4*z).

%F a(n) = Sum_{k=0..n} C(n+k, 2k) * 2^k * C(k), C(n) given by A000108. - _Paul Barry_, May 21 2005

%F a(n) = Sum_{k=0..n} A060693(n,k)*2^(n-k). - _Philippe Deléham_, Apr 02 2007

%F a(0) = 1, a(n) = a(n-1) + 2*Sum_{k=0..n-1} a(k)*a(n-1-k). - _Philippe Deléham_, Oct 23 2007

%F a(n) = (3/2)*A107841(n) for n > 0. - _Philippe Deléham_, Oct 28 2007

%F G.f.: 1/(1-x-2*x/(1-x-2*x/(1-x-2*x/(1-.... (continued fraction). - _Paul Barry_, Feb 01 2009

%F G.f.: 1/(1-3*x-6*x^2/(1-5*x-6*x^2/(1-5*x-6*x^2/(1-... (continued fraction). - _Paul Barry_, Apr 28 2009

%F G.f.: 1/(1-3*x/(1-2*x/(1-3*x/(1-2*x/(1-3*x/(1-... (continued fraction). - _Paul Barry_, May 14 2009

%F a(n) = Hypergeometric2F1(-n,n+1;2;-2) = Sum_{k=0..n} C(n+k,k) * C(n,k) * 2^k/(k+1). - _Paul Barry_, Feb 08 2011

%F G.f.: A(x) = (1-x-(x^2-10*x+1)^(1/2))/(4*x) = 1/(G(0)-x); G(k)= 1 + x - 3*x/G(k+1); (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Jan 05 2012

%F D-finite with recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - (n-2)*a(n-2). - _Vaclav Kotesovec_, Oct 17 2012

%F a(n) ~ sqrt(12+5*sqrt(6))*(5+2*sqrt(6))^n/(4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 17 2012

%F G.f. A(x) satisfies: A(x) = (1 + 2*x*A(x)^2) / (1 - x). - _Ilya Gutkovskiy_, Jun 30 2020

%p A103210 := proc(n)

%p if n = 0 then

%p 1;

%p else

%p add(binomial(n,i)*binomial(n,i+1)*2^i*3^(n-i),i=0..n-1)/n ;

%p end if;

%p end proc: # _R. J. Mathar_, Feb 10 2015

%p A103210_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;

%p for w from 1 to n do a[w] := 3*a[w-1] + 2*add(a[j]*a[w-j-1], j=1..w-1) od;

%p convert(a, list) end: A103210_list(21); # _Peter Luschny_, Feb 29 2016

%t CoefficientList[Series[(1-x-Sqrt[x^2-10*x+1])/(4*x), {x, 0, 25}], x] (* _Vaclav Kotesovec_, Oct 17 2012 *)

%t A103210[n_]:= Hypergeometric2F1[-n, n+1, 2, -2]; Table[A103210[n], {n, 0, 25}] (* _Peter Luschny_, Jan 07 2018 *)

%o (PARI) my(x='x+O('x^25)); Vec((1-x-sqrt(x^2-10*x+1))/(4*x)) \\ _G. C. Greubel_, Feb 10 2018

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!((1-x-Sqrt(x^2-10*x+1))/(4*x))); // _G. C. Greubel_, Feb 10 2018

%o (Sage) [1]+[(3^n/n)*sum( binomial(n,j)*binomial(n,j+1)*(2/3)^j for j in (0..n-1)) for n in (1..25)] # _G. C. Greubel_, Jun 08 2020

%Y Third column of array A103209.

%Y Cf. A000108, A060693, A081955, A107841.

%K nonn

%O 0,2

%A _Ralf Stephan_, Jan 27 2005

%E Spelling/notation corrections by _Charles R Greathouse IV_, Mar 18 2010