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%I #43 Mar 21 2024 08:32:02
%S 1,2,3,6,11,24,48,110,231,546,1183,2856,6324,15504,34884,86526,197087,
%T 493350,1134705,2861430,6633315,16829280,39268320,100134216,234930276,
%U 601661144,1418201268,3645533040,8627761528,22249511328
%N If n mod 2 = 0 then m := n/2 and a(n) = (3*m)!*(5*m+1)/((m+1)!*(2*m+1)!); otherwise m := (n-1)/2, a(n) = 6*(3*m+2)!/(m!*(2*m+3)!).
%H L. W. Beineke and R. E. Pippert, <a href="http://dx.doi.org/10.4153/CJM-1974-006-x">Enumerating dissectable polyhedra by their automorphism groups</a>, Canad. J. Math., 26 (1974), 50-67.
%F From _Gary W. Adamson_, Jul 14 2011: (Start)
%F a(n) = sum of top row terms in M^n, M = the infinite square production matrix:
%F 1, 1, 0, 0, 0, 0, ...
%F 0, 0, 1, 0, 0, 0, ...
%F 1, 1, 0, 1, 0, 0, ...
%F 0, 0, 1, 0, 1, 0, ...
%F 1, 1, 0, 1, 0, 1, ...
%F ... (End)
%F 8*n*(n+2)*a(n) + 4*(7*n^2 - 7*n - 17)*a(n-1) + 6*(-9*n^2 + 9*n - 17)*a(n-2) - 21*(3*n-5)*(3*n-7)*a(n-3) = 0. - _R. J. Mathar_, Jul 10 2013
%F From _Robert A. Russell_, Mar 20 2024: (Start)
%F a(n) = V(n) in the Beineke and Pippert link.
%F G.f.: 2*(G(z^2) - 1)/z + 2*G(z^2)^2 - G(z^2), where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)
%p series(RootOf(x*A^3-2*A^2+3*A-1, A)^2, x=0, 30); # _Mark van Hoeij_, May 16 2013
%t a[0] = 1; a[1] = 2; a[n_] := a[n] = 3(2n+3)(3n-4)(3n-2)a[n-2]/(4n(n+2)(2n+1)) + (3(18n+16)a[n-1])/(4n(n+2)(2n+1)); Table[a[n], {n, 0, 29}] (* _Jean-François Alcover_, Dec 02 2016 *)
%t Table[If[OddQ[n],6Binomial[(3n+1)/2,n+1]/(n+2),(5n+2)Binomial[3n/2,n/2] / ((n+1)(n+2))],{n,0,30}] (* _Robert A. Russell_, Feb 16 2024 *)
%o (PARI) a047750(n)={if(n%2,my(m=(n-1)/2);6*(3*m+2)!/(m!*(2*m+3)!),my(m=n/2);(3*m)!*(5*m+1)/((m+1)!*(2*m+1)!))};
%o for(k=0,29,print1(a047750(k),", ")) \\ _Hugo Pfoertner_, Mar 07 2020
%Y Cf. A001764, A047749, A047760, A047773.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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