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

1, 2, 8, 43, 268, 1824, 13156, 98865, 765948, 6075256, 49094708, 402801425, 3346590068, 28099903160, 238079915640, 2032914717645, 17476713955548, 151143219598008, 1314045772469632, 11478299163026540, 100688538612524720, 886622619082002120, 7834289222109530340

Offset: 0

Author

Valery A. Liskovets

Keywords

• nonn

Comments

The number of rooted isthmusless n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005

References

• L. M. Koganov, V. A. Liskovets, T. R. S. Walsh, Total vertex enumeration in rooted planar maps, Ars Combin. 54 (2000), 149-160.
• V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Crossrefs

Cf. A000260, A005470, A002293, A342981.

Programs

Maple

12*n*(4*n-1)!*(5*n^2+13*n+2)/(n!*(3*n+3)!);

Mathematica

Join[{1},Table[12n (4n-1)! (5n^2+13n+2)/(n!(3n+3)!),{n,20}]] (* Harvey P. Dale, May 20 2018 *)

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

Formula

a(n) = 12*n*(4*n-1)!*(5*n^2+13*n+2)/(n!*(3*n+3)!) for n > 0.

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

a(n) = Sum_{k=1..n+1} k*A342981(n, k). - Andrew Howroyd, Apr 06 2021

Extensions

Offset corrected and terms a(21) and beyond from Andrew Howroyd, Apr 06 2021