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Number of loopless rooted planar maps with 4 faces and n vertices and no isthmuses.
(Formerly M5091)
2

%I M5091 #30 Jul 18 2024 08:51:31

%S 1,20,131,469,1262,2862,5780,10725,18647,30784,48713,74405,110284,

%T 159290,224946,311429,423645,567308,749023,976373,1258010,1603750,

%U 2024672,2533221,3143315,3870456,4731845,5746501,6935384,8321522,9930142,11788805,13927545,16379012

%N Number of loopless rooted planar maps with 4 faces and n vertices and no isthmuses.

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

%H Andrew Howroyd, <a href="/A006417/b006417.txt">Table of n, a(n) for n = 2..1000</a>

%H T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(75)90050-7">Counting rooted maps by genus. III: Nonseparable maps</a>, J. Combinatorial Theory Ser. B 18 (1975), 222-259.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F From _Colin Barker_, Apr 08 2013: (Start)

%F a(n) = (2160+846*n-1027*n^2-345*n^3+125*n^4+39*n^5+2*n^6)/360.

%F G.f.: -x^2*(5*x^6-29*x^5+65*x^4-63*x^3+12*x^2+13*x+1) / (x-1)^7. (End)

%F E.g.f.: exp(x)*(2160 - 360*x - 540*x^2 + 1560*x^3 + 645*x^4 + 69*x^5 + 2*x^6)/360 - 6 - 5*x. - _Stefano Spezia_, Jul 18 2024

%o (PARI) a(n)={if(n<2, 0, (2*n^6 + 39*n^5 + 125*n^4 - 345*n^3 - 1027*n^2 + 846*n + 2160)/360)} \\ _Andrew Howroyd_, Apr 01 2021

%Y Column k=4 of A342980.

%K nonn,easy

%O 2,2

%A _N. J. A. Sloane_

%E Title improved by _Sean A. Irvine_, Apr 03 2017

%E Terms a(14) and beyond from _Andrew Howroyd_, Apr 01 2021