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 A005775 Number of compact-rooted directed animals of size n having 3 source points. (Formerly M3481) 5
 1, 4, 14, 45, 140, 427, 1288, 3858, 11505, 34210, 101530, 300950, 891345, 2638650, 7809000, 23107488, 68375547, 202336092, 598817490, 1772479905, 5247421410, 15538054455, 46019183840, 136325212750, 403933918375, 1197131976846, 3548715207534, 10521965227669 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,2 COMMENTS Binomial transform of A037955. - Paul Barry, Dec 28 2006 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 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 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Reinhard Zumkeller, Table of n, a(n) for n = 3..1000 D. Gouyou-Beauchamps, G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357. FORMULA (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 G.f.: (x^2 + x - 1 +(x^2 - 3*x + 1)*sqrt((1+x)/(1-3*x)))/(2*x^2). E.g.f.: exp(x)*(Bessel_I(2,2*x) + Bessel_I(3,2*x)); a(n+1)=sum{k=0..n, C(n,k)*C(k,floor(k/2)-1)};. - Paul Barry, Dec 28 2006 a(n) ~ 3^(n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 25 2014 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 a(n) = GegenbauerC(n-4,-n+1,-1/2) + GegenbauerC(n-3,-n+1,-1/2). - Peter Luschny, May 12 2016 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 a(n) = A005773(n) - A001006(n) for n>=3. - John Keith, Nov 20 2020 EXAMPLE 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 + ... MAPLE seq(simplify(GegenbauerC(n-4, -n+1, -1/2) + GegenbauerC(n-3, -n+1, -1/2)), n=3..28); # Peter Luschny, May 12 2016 MATHEMATICA 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 *) PROG (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)}; (PARI) {a(n) = n--; sum(k=0, n, binomial(n, k) * binomial(k, k\2 -1))}; /* Michael Somos, May 12 2016 */ (Haskell) a005775 = flip a038622 2 . (subtract 1)  -- Reinhard Zumkeller, Feb 26 2013 CROSSREFS Cf. A005773. k=2 column of array in A038622. Cf. A005774, A066822. Sequence in context: A182902 A108765 A304068 * A094688 A068092 A255678 Adjacent sequences:  A005772 A005773 A005774 * A005776 A005777 A005778 KEYWORD nonn,easy,changed AUTHOR EXTENSIONS More terms from Randall L. Rathbun, Jan 19 2002 Edited by Michael Somos, Feb 02 2002 STATUS approved

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Last modified December 5 19:53 EST 2020. Contains 338965 sequences. (Running on oeis4.)