A172481
a(n) = (3*n*2^n+2^(n+4)+2*(-1)^n)/18.
Original entry on oeis.org
1, 2, 5, 11, 25, 55, 121, 263, 569, 1223, 2617, 5575, 11833, 25031, 52793, 111047, 233017, 487879, 1019449, 2126279, 4427321, 9204167, 19107385, 39612871, 82021945, 169636295, 350457401, 723284423, 1491308089, 3072094663, 6323146297, 13004206535, 26724240953
Offset: 0
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[(3*n*2^n+2^(n+4)+2*(-1)^n)/18: n in [0..40]]; // Vincenzo Librandi, Aug 04 2011
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Table[(3n 2^n+2^(n+4)+2(-1)^n)/18,{n,0,40}] (* or *)
CoefficientList[Series[(1-x-x^2)/((1+x)(1-2x)^2), {x,0,40}], x] (* Harvey P. Dale, Mar 28 2011 *)
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a(n)=(3*n*2^n+2^(n+4)+2*(-1)^n)/18 \\ Charles R Greathouse IV, Oct 07 2015
Definition replaced by explicit formula by
R. J. Mathar, Feb 11 2010
A126183
Triangle read by rows: T(n,k) is number of hex trees with n edges and k nonroot nodes of outdegree 2.
Original entry on oeis.org
1, 3, 10, 33, 3, 108, 29, 351, 186, 6, 1134, 990, 95, 3645, 4725, 900, 15, 11664, 20979, 6615, 329, 37179, 88452, 41580, 4116, 42, 118098, 358668, 234738, 38556, 1176, 373977, 1410615, 1224720, 300510, 18270, 126, 1180980, 5412825, 6014250
Offset: 0
Triangle begins:
1;
3;
10;
33, 3;
108, 29;
351, 186, 6;
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G := 1/2/t^2/z^2*(-11*t*z^2+2*t^2*z^2+3*z*t+9*z^2-6*z+1-sqrt(1-58*t*z^2-12*z+54*z^2 +6*z*t+81*z^4-108*z^3 -36*t^3*z^4+153*t^2*z^4 -198*t*z^4-78*t^2*z^3+186*t*z^3+9*t^2*z^2)): Gser:=simplify(series(G,z=0,16)): for n from 0 to 18 do P[n]:=sort(coeff(Gser,z,n)) od: 1; for n from 1 to 13 do seq(coeff(P[n],t,j),j=0..floor((n-1)/2)) od; # yields sequence in triangular form
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len = 40; m = Ceiling[2 Sqrt[len]]; gf[t_, z_] = g /. Solve[g == 1 + 3z* h + z^2*h^2 && h == 1 + 3z*h + t*z^2*h^2, g, h][[1]]; gser = Series[gf[t, z], {z, 0, m}]; p[n_] := Coefficient[gser, z, n]; tr[n_, k_] := tr[n, k] = Coefficient[p[n], t, k]; Flatten[Table[ tr[n, k], {n, 0, m}, {k, 0, Max[0, Floor[(n-1)/2]]}]][[1 ;; len]] (* Jean-François Alcover, May 31 2011, after Maple prog. *)
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