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A027836 Total number of vertices in all loopless rooted planar maps with n edges.

%I A027836 #22 Apr 06 2021 20:15:24
%S A027836 1,2,8,43,268,1824,13156,98865,765948,6075256,49094708,402801425,
%T A027836 3346590068,28099903160,238079915640,2032914717645,17476713955548,
%U A027836 151143219598008,1314045772469632,11478299163026540,100688538612524720,886622619082002120,7834289222109530340
%N A027836 Total number of vertices in all loopless rooted planar maps with n edges.
%C A027836 The number of rooted isthmusless n-edge maps in the plane (planar with a distinguished outside face). - _Valery A. Liskovets_, Mar 17 2005
%D A027836 L. M. Koganov, V. A. Liskovets, T. R. S. Walsh, Total vertex enumeration in rooted planar maps, Ars Combin. 54 (2000), 149-160.
%D A027836 V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
%H A027836 Andrew Howroyd, <a href="/A027836/b027836.txt">Table of n, a(n) for n = 0..500</a>
%H A027836 V. A. Liskovets and T. R. Walsh, <a href="http://dx.doi.org/10.1016/j.aam.2005.03.006">Counting unrooted maps on the plane</a>, Advances in Applied Math., 36, No.4 (2006), 364-387.
%F A027836 a(n) = 12*n*(4*n-1)!*(5*n^2+13*n+2)/(n!*(3*n+3)!) for n > 0.
%F A027836 G.f.: -(1-3*g+g^2)*g where g = 1+x*g^4 is the g.f. of A002293. - _Mark van Hoeij_, Nov 11 2011
%F A027836 a(n) = Sum_{k=1..n+1} k*A342981(n, k). - _Andrew Howroyd_, Apr 06 2021
%p A027836 12*n*(4*n-1)!*(5*n^2+13*n+2)/(n!*(3*n+3)!);
%t A027836 Join[{1},Table[12n (4n-1)! (5n^2+13n+2)/(n!(3n+3)!),{n,20}]] (* _Harvey P. Dale_, May 20 2018 *)
%o A027836 (PARI) a(n) = if(n==0, 1, 12*n*(4*n-1)!*(5*n^2+13*n+2)/(n!*(3*n+3)!)) \\ _Andrew Howroyd_, Apr 06 2021
%Y A027836 Cf. A000260, A005470, A002293, A342981.
%K A027836 nonn
%O A027836 0,2
%A A027836 _Valery A. Liskovets_
%E A027836 Offset corrected and terms a(21) and beyond from _Andrew Howroyd_, Apr 06 2021