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A103211 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*3^i*4^(n-i), a(0)=1.

%I A103211 #86 May 04 2025 03:25:38
%S A103211 1,4,28,244,2380,24868,272188,3080596,35758828,423373636,5092965724,
%T A103211 62071299892,764811509644,9511373563492,119231457692284,
%U A103211 1505021128450516,19112961439180588,244028820862442116,3130592301487969948,40333745806536135028,521655330655122923980
%N A103211 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*3^i*4^(n-i), a(0)=1.
%C A103211 The Hankel transform of this sequence is 12^C(n+1,2). - _Philippe Deléham_, Oct 28 2007
%C A103211 The sequence 1, 1, 4, 28, ... has a(n) = 0^n + Sum_{k=0..n-1} C(n+k-1, 2*k)*C(k)*3^k and Hankel transform 3^C(n+1, 2)*4^C(n, 2). - _Paul Barry_, Dec 09 2008
%C A103211 Number of Dyck n-paths with two colors of up (U,u) and two colors of down (D,d) avoiding DU. - _David Scambler_, Jun 24 2013
%H A103211 Vincenzo Librandi, <a href="/A103211/b103211.txt">Table of n, a(n) for n = 0..200</a>
%H A103211 J. Abate and W. 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(3).
%H A103211 E. Ackerman, G. Barequet, R. Y. Pinter and D. Romik, <a href="http://dx.doi.org/10.1016/j.ipl.2006.01.011">The number of guillotine partitions in d dimensions</a>, Inf. Proc. Lett. 98 (4) (2006) 162-167
%H A103211 Paul Barry, <a href="http://arxiv.org/abs/1312.0583">Embedding structures associated with Riordan arrays and moment matrices</a>, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
%H A103211 Z. Chen and H. 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=4, b=3.
%H A103211 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 A103211 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 A103211 Djamila Oudrar, <a href="http://arxiv.org/abs/1604.05839">Sur l'énumération de structures discrètes, une approche par la théorie des relations</a>, Thesis (in French), arXiv:1604.05839 [math.CO], 2016.
%F A103211 G.f.: (1-z-sqrt(z^2-14*z+1))/(6*z).
%F A103211 a(n) = Sum_{k=0..n} C(n+k,2k)*3^k*C(k), C(n) given by A000108. - _Paul Barry_, May 21 2005
%F A103211 a(n) = Sum_{k=0..n} A060693(n,k)*3^(n-k). - _Philippe Deléham_, Apr 02 2007
%F A103211 a(0)=1, a(n) = a(n-1) + 3*Sum_{k=0..n-1} a(k)*a(n-1-k). - _Philippe Deléham_, Oct 23 2007
%F A103211 G.f.: 1/(1-x-3*x/(1-x-3*x/(1-x-3*x/(1-x-3*x/(1-... (continued fraction). - _Paul Barry_, Nov 07 2009
%F A103211 D-finite with recurrence: (n+1)*a(n) = 7*(2*n-1)*a(n-1) - (n-2)*a(n-2). - _Vaclav Kotesovec_, Oct 17 2012
%F A103211 a(n) ~ sqrt(24+14*sqrt(3))*(7+4*sqrt(3))^n/(6*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 17 2012
%F A103211 a(n) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k)*hypergeom([k - n, n + 1], [k + 2], 4). - _Peter Luschny_, Jan 08 2018
%F A103211 G.f. A(x) satisfies: A(x) = (1 + 3*x*A(x)^2) / (1 - x). - _Ilya Gutkovskiy_, Jun 30 2020
%F A103211 From _Michael Somos_, Mar 15 2024: (Start)
%F A103211 a(n) = (4/3)*A131763(n) for n>0.
%F A103211 Given g.f. A(x) and y = -x*A(-x^2), then 3*y-1/y = x+1/x.
%F A103211 If a(n) := a(-1-n) for n<0, then 0 = a(n)*(+a(n+1) -35*a(n+2) +4*a(n+3)) +a(n+1)*(+7*a(n+1) +194*a(n+2) -35*a(n+3)) +a(n+2)*(+7*a(n+2) +a(n+3)) for all n in Z. (End)
%e A103211 G.f. = 1 + 4*x + 28*x^2 + 244*x^3 + 2380*x^4 + 24868*x^5 + ... _Michael Somos_, Mar 15 2024
%p A103211 A103211_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p A103211 for w from 1 to n do a[w] := a[w-1] + 3*add(a[j]*a[w-j-1], j=0..w-1) od;
%p A103211 convert(a, list) end: A103211_list(20); # _Peter Luschny_, Feb 29 2016
%t A103211 CoefficientList[Series[(1-x-Sqrt[x^2-14*x+1])/(6*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%t A103211 a[n_] := Sum[(-1)^(n - k) Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, 4], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* _Peter Luschny_, Jan 08 2018 *)
%t A103211 a[ n_] := If[n < 0, a[-1-n], SeriesCoefficient[2/(1 - x + Sqrt[1 - 14*x + x^2]), {x, 0, n}]]; (* _Michael Somos_, Mar 15 2024 *)
%o A103211 (PARI) x='x+O('x^30); Vec((1-x-sqrt(x^2-14*x+1))/(6*x)) \\ _G. C. Greubel_, Feb 10 2018
%o A103211 (PARI) {a(n) = if(n<0, a(-1-n), polcoeff(2/(1 - x + sqrt(1 - 14*x + x^2 + x*O(x^n))), n))}; /* _Michael Somos_, Mar 15 2024 */
%o A103211 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-14*x+1))/(6*x))) // _G. C. Greubel_, Feb 10 2018
%o A103211 (GAP) a:=n->(1/n)*Sum([0..n-1],i->Binomial(n,i)*Binomial(n,i+1)*
%o A103211 3^i*4^(n-i));;
%o A103211 A103211:=Concatenation([1],List([1..20],n->a(n))); # _Muniru A Asiru_, Feb 11 2018
%Y A103211 Fourth column of array A103209.
%Y A103211 Cf. A131763.
%K A103211 nonn
%O A103211 0,2
%A A103211 _Ralf Stephan_, Jan 27 2005