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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. 15
 1, 4, 28, 244, 2380, 24868, 272188, 3080596, 35758828, 423373636, 5092965724, 62071299892, 764811509644, 9511373563492, 119231457692284, 1505021128450516, 19112961439180588, 244028820862442116, 3130592301487969948, 40333745806536135028, 521655330655122923980 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The Hankel transform of this sequence is 12^C(n+1,2). - Philippe Deléham, Oct 28 2007 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 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 J. Abate, W. Whitt, Integer Sequences from Queueing Theory , J. Int. Seq. 13 (2010), 10.5.5, b_n(3). E. Ackerman, G. Barequet, R. Y. Pinter and D. Romik, The number of guillotine partitions in d dimensions, Inf. Proc. Lett. 98 (4) (2006) 162-167 P. Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013. Z. Chen, H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], (2016), eq. (1.13), a=4, b=3. Samuele Giraudo, Operads from posets and Koszul duality, arXiv preprint arXiv:1504.04529 [math.CO], 2015. Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016. Djamila Oudrar, Sur l'énumération de structures discrètes, une approche par la théorie des relations, Thesis (in French), arXiv:1604.05839 [math.CO], 2016. FORMULA G.f.: (1-z-sqrt(z^2-14*z+1))/(6*z). a(n) = Sum_{k=0..n} C(n+k,2k)*3^k*C(k), C(n) given by A000108. - Paul Barry, May 21 2005 a(n) = Sum_{k=0..n} A060693(n,k)*3^(n-k). - Philippe Deléham, Apr 02 2007 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 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 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 a(n) ~ sqrt(24+14*sqrt(3))*(7+4*sqrt(3))^n/(6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012 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 G.f. A(x) satisfies: A(x) = (1 + 3*x*A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jun 30 2020 MAPLE A103211_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1; 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; convert(a, list) end: A103211_list(20); # Peter Luschny, Feb 29 2016 MATHEMATICA CoefficientList[Series[(1-x-Sqrt[x^2-14*x+1])/(6*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *) 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 *) PROG (PARI) x='x+O('x^30); Vec((1-x-sqrt(x^2-14*x+1))/(6*x)) \\ G. C. Greubel, Feb 10 2018 (MAGMA) Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-14*x+1))/(6*x))) // G. C. Greubel, Feb 10 2018 (GAP) a:=n->(1/n)*Sum([0..n-1], i->Binomial(n, i)*Binomial(n, i+1)* 3^i*4^(n-i));; A103211:=Concatenation([1], List([1..20], n->a(n))); # Muniru A Asiru, Feb 11 2018 CROSSREFS Fourth column of array A103209. Sequence in context: A188266 A192625 A199561 * A229644 A354602 A228714 Adjacent sequences:  A103208 A103209 A103210 * A103212 A103213 A103214 KEYWORD nonn AUTHOR Ralf Stephan, Jan 27 2005 STATUS approved

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Last modified August 9 00:15 EDT 2022. Contains 356016 sequences. (Running on oeis4.)