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Number of genus 5 rooted hypermaps with n darts.
2

%I #14 Aug 23 2022 14:11:54

%S 604800,57170880,2936606400,108502598960,3225186125460,81861294718764,

%T 1840409325096500,37558997857897164,708015469597497732,

%U 12488421105878928700,208161512148250424484,3304395638081490531324,50267199680265668419244,736516493829967530909204,10437808798822929984593100

%N Number of genus 5 rooted hypermaps with n darts.

%H Gheorghe Coserea, <a href="/A321705/b321705.txt">Table of n, a(n) for n = 11..111</a>

%H Mednykh, A.; Nedela, R. <a href="https://doi.org/10.1007/s10958-017-3555-5">Recent progress in enumeration of hypermaps</a>, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 7

%H Peter Zograf, <a href="https://arxiv.org/abs/1312.2538">Enumeration of Grothendieck's Dessins and KP Hierarchy</a>, arXiv:1312.2538 [math.CO], 2014.

%F G.f.: y*(y - 1)^11*(13150*y^19 - 315600*y^18 + 6947865*y^17 - 70489470*y^16 + 569637816*y^15 - 3253135788*y^14 + 14658702716*y^13 - 51696766668*y^12 + 146255446788*y^11 - 332779761068*y^10 + 610739916966*y^9 - 900544355928*y^8 + 1057440629016*y^7 - 973453624356*y^6 + 685359139356*y^5 - 355019010868*y^4 + 127180243662*y^3 - 28342783668*y^2 + 3224985513*y - 120590634)/(4*(y - 2)^22*(y + 1)^17), where y=A000108(2*x).

%o (PARI)

%o seq(N) = {

%o my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));

%o Vec(y*(y - 1)^11*(13150*y^19 - 315600*y^18 + 6947865*y^17 - 70489470*y^16 + 569637816*y^15 - 3253135788*y^14 + 14658702716*y^13 - 51696766668*y^12 + 146255446788*y^11 - 332779761068*y^10 + 610739916966*y^9 - 900544355928*y^8 + 1057440629016*y^7 - 973453624356*y^6 + 685359139356*y^5 - 355019010868*y^4 + 127180243662*y^3 - 28342783668*y^2 + 3224985513*y - 120590634)/(4*(y - 2)^22*(y + 1)^17));

%o };

%o seq(15)

%Y Column 5 of A321710.

%K nonn

%O 11,1

%A _Gheorghe Coserea_, Nov 17 2018