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A305873 Coefficients of polynomials g_b(x) that arise in the generating function for rooted maps (A053979).

%I A305873 #13 Jun 02 2025 15:21:38
%S A305873 1,3,5,15,65,60,105,804,1730,1105,945,10824,39110,55645,27120,10395,
%T A305873 162357,854250,1987270,2105070,828250,135135,2714445,19180410,
%U A305873 63897550,108878610,91692550,30220800,2027025,50301360,452984532,2004435096,4836052370,6479714440,4523710100,1282031525
%N A305873 Coefficients of polynomials g_b(x) that arise in the generating function for rooted maps (A053979).
%C A305873 The generating function of the b-th subdiagonal of A053979 is g_b(y)*(1-sqrt(1-4x))/2/(1-4x)^b, b>=0, where g_b(y) = 1 (b= 0 or 1), 3+5*y (b=2), 15+65*y+60*y^2 (b=3) etc are the coefficients in this table, and where y=(1/sqrt(1-4x)-1)/2.
%C A305873 The coefficient 794 cited by Walsh-Lehman (1972) has been corrected to 804.
%H A305873 T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(72)90056-1">Counting rooted maps by genus. I</a>, J. Comb. Theory B 13 (1972), 192-218, eq. (5a).
%p A305873 A305873:= proc(b,x)
%p A305873     local gn1,k ;
%p A305873     option remember;
%p A305873     if b = 0 or b= 1 then
%p A305873         return 1 ;
%p A305873     else
%p A305873         gn1 := procname(b-1,x) ;
%p A305873         add(procname(k,x)*procname(b-k,x),k=1..b-1) ;
%p A305873         gbx := %*x+(2*(b-1)*(1+2*x)+1)*gn1 ;
%p A305873         expand(gbx+2*x*(x+1)*diff(gn1,x)) ;
%p A305873     end if;
%p A305873 end proc:
%p A305873 for b from 1 to 8 do
%p A305873     gx := A305873(b,x) ;
%p A305873     for l from 0 to b-1 do
%p A305873         printf("%d,",coeff(gx,x,l)) ;
%p A305873     end do:
%p A305873     printf("\n") ;
%p A305873 end do:
%t A305873 A305873[b_, x_] := A305873[b, x] = Module[{gn1, k, s}, If[b == 0 || b == 1, Return@1, gn1 = A305873[b - 1, x]; s = Sum[A305873[k, x]*A305873[b - k, x], {k, 1, b - 1}]; gbx = s*x + (2*(b - 1)*(1 + 2*x) + 1)*gn1; Expand[gbx + 2*x*(x + 1)*D[gn1, x]]]];
%t A305873 Reap[For[b = 1, b <= 8, b++, gx = A305873[b, x]; For[l = 0, l <= b - 1, l++, Sow[Coefficient[gx, x, l]]]]][[2, 1]] (* _Jean-François Alcover_, Nov 09 2023, after Maple program *)
%Y A305873 Cf. A062980 (diagonal), A001147 (first column)
%K A305873 nonn,tabl,easy
%O A305873 1,2
%A A305873 _R. J. Mathar_, Jun 12 2018