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Number of certain rooted planar maps.
(Formerly M3543 N1435)
2

%I M3543 N1435 #30 Aug 21 2016 01:57:52

%S 1,4,18,89,466,2537,14209,81316,473338,2793454,16674417,100487896,

%T 610549829,3735850007,23000055178,142370597601,885521350882,

%U 5531501612071,34686798239678,218273864005214,1377897874711437

%N Number of certain rooted planar maps.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A000305/b000305.txt">Table of n, a(n) for n = 1..200</a>

%H W. G. Brown, <a href="http://dx.doi.org/10.4153/CJM-1963-056-7">Enumeration of non-separable planar maps</a>, Canad. J. Math., 15 (1963), 526-545.

%H W. G. Brown, <a href="/A000087/a000087.pdf">Enumeration of non-separable planar maps</a> [Annotated scanned copy]

%p with(linalg): T := proc(n,k) if k<=n then k*sum((2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/(2*k-j-1)!/(n-j)!,j=k..min(n,2*k-1))/(2*n-k+1)! else 0 fi end:A := matrix(30,30,T): seq(sum(A[i,j],j=1..i),i=1..30);

%p R := RootOf(x-t*(t-1)^2, t); ogf := series((R+1)/((1-R-R^2)*(R-1)^2), x=0, 20); # _Mark van Hoeij_, Nov 08 2011

%t t[n_, k_] := If[k <= n, k*Sum[(2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/ (2*k-j-1)!/(n-j)!, {j, k, Min[n, 2*k-1]}]/(2*n-k+1)!, 0]; a[n_] := Sum[ t[n, k], {k, 1, n}]; Array[a, 21] (* _Jean-François Alcover_, Feb 07 2016 after Herman Jamke in A046652 *)

%Y Row sums of A046652.

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Emeric Deutsch_, Mar 03 2004