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%I M3481 #74 Mar 07 2023 02:32:53
%S 1,4,14,45,140,427,1288,3858,11505,34210,101530,300950,891345,2638650,
%T 7809000,23107488,68375547,202336092,598817490,1772479905,5247421410,
%U 15538054455,46019183840,136325212750,403933918375,1197131976846,3548715207534,10521965227669
%N Number of compact-rooted directed animals of size n having 3 source points.
%C Binomial transform of A037955. - _Paul Barry_, Dec 28 2006
%C Apparently, the number of Dyck paths of semilength n that contain at least one UUU but avoid UUU's starting above level 0. - _David Scambler_, Jul 02 2013
%C a(n) = number of paths in the half-plane x >= 0 from (0,0) to (n-1,2) or (n-1,-3), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=5, we have the 14 paths: HHUU, UUHH, UHHU, HUUH, HUHU, UHUH, UDUU, UUDU, UUUD, DUUU, DDDH, HDDD, DHDD, DDHD. - _José Luis Ramírez Ramírez_, Apr 19 2015
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A005775/b005775.txt">Table of n, a(n) for n = 3..1000</a>
%H Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H D. Gouyou-Beauchamps and G. Viennot, <a href="http://dx.doi.org/10.1016/0196-8858(88)90017-6">Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem</a>, Adv. in Appl. Math. 9 (1988), no. 3, 334-357.
%F D-finite with recurrence (n+2)*(n-3)*a(n) = 2*n*(n-1)*a(n-1) + 3*(n-1)*(n-2)*a(n-2), a(2)=0, a(3)=1. - _Michael Somos_, Feb 02 2002
%F G.f.: (x^2 + x - 1 +(x^2 - 3*x + 1)*sqrt((1+x)/(1-3*x)))/(2*x^2).
%F From _Paul Barry_, Dec 28 2006: (Start)
%F E.g.f.: exp(x)*(Bessel_I(2,2*x) + Bessel_I(3,2*x));
%F a(n+1) = Sum_{k=0..n} C(n,k)*C(k,floor(k/2)-1). (End)
%F a(n) ~ 3^(n-1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Feb 25 2014
%F G.f.: (z^3*M(z)^2+z^4*M(z)^3)/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths. - _José Luis Ramírez Ramírez_, Apr 19 2015
%F a(n) = GegenbauerC(n-4,-n+1,-1/2) + GegenbauerC(n-3,-n+1,-1/2). - _Peter Luschny_, May 12 2016
%F 0 = a(n)*(+9*a(n+1) - 63*a(n+2) - 54*a(n+3) + 87*a(n+4) - 21*a(n+5))+ a(n+1)*(+21*a(n+1) + 79*a(n+2) + 13*a(n+3) - 118*a(n+4) + 35*a(n+5)) + a(n+2)*(-14*a(n+2) + 79*a(n+3) - 67*a(n+4) + 14*a(n+5)) + a(n+3)*(+6*a(n+3) + 19*a(n+4) - 11*a(n+5)) + a(n+4)*(+a(n+4) + a(n+5)) if n >= 0. - _Michael Somos_, May 12 2016
%F a(n) = A005773(n) - A001006(n) for n >= 3. - _John Keith_, Nov 20 2020
%e G.f. = x^3 + 4*x^4 + 14*x^5 + 45*x^6 + 140*x^7 + 427*x^8 + 1288*x^9 + 3858*x^10 + ...
%p seq(simplify(GegenbauerC(n-4,-n+1,-1/2) + GegenbauerC(n-3,-n+1,-1/2)),n=3..28); # _Peter Luschny_, May 12 2016
%t nmax = 28; t[n_ /; n > 0, k_ /; k >= 1] := t[n, k] = t[n-1, k-1] + t[n-1, k] + t[n-1, k+1]; t[0, 0] = 1; t[0, _] = 0; t[_?Negative, _?Negative] = 0; t[n_, 0] := 2*t[n-1, 0] + t[n-1, 1]; a[n_] := t[n-1, 2]; Table[a[n], {n, 3, nmax} ] (* _Jean-François Alcover_, Jul 03 2013, from A038622 *)
%o (PARI) {a(n) = polcoeff( (x^2 + x - 1 + (x^2 - 3*x + 1) * sqrt((1 + x) / (1 - 3*x) + x^3 * O(x^n))) / (2*x^2), n)};
%o (PARI) {a(n) = n--; sum(k=0, n, binomial(n, k) * binomial(k, k\2 -1))}; /* _Michael Somos_, May 12 2016 */
%o (Haskell)
%o a005775 = flip a038622 2 . (subtract 1) -- _Reinhard Zumkeller_, Feb 26 2013
%Y Cf. A005773.
%Y k=2 column of array in A038622.
%Y Cf. A005774, A066822.
%K nonn,easy
%O 3,2
%A _Simon Plouffe_
%E More terms from _Randall L Rathbun_, Jan 19 2002
%E Edited by _Michael Somos_, Feb 02 2002
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