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

1, 3, 5, 15, 65, 60, 105, 804, 1730, 1105, 945, 10824, 39110, 55645, 27120, 10395, 162357, 854250, 1987270, 2105070, 828250, 135135, 2714445, 19180410, 63897550, 108878610, 91692550, 30220800, 2027025, 50301360, 452984532, 2004435096, 4836052370, 6479714440, 4523710100, 1282031525

Offset: 1

Author

R. J. Mathar, Jun 12 2018

Keywords

• nonn
• tabl
• easy

Comments

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.

The coefficient 794 cited by Walsh-Lehman (1972) has been corrected to 804.

Links

• T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. I, J. Comb. Theory B 13 (1972), 192-218, eq. (5a).

Crossrefs

Cf. A062980 (diagonal), A001147 (first column)

Programs

Maple

A305873:= proc(b,x)
    local gn1,k ;
    option remember;
    if b = 0 or b= 1 then
        return 1 ;
    else
        gn1 := procname(b-1,x) ;
        add(procname(k,x)*procname(b-k,x),k=1..b-1) ;
        gbx := %*x+(2*(b-1)*(1+2*x)+1)*gn1 ;
        expand(gbx+2*x*(x+1)*diff(gn1,x)) ;
    end if;
end proc:
for b from 1 to 8 do
    gx := A305873(b,x) ;
    for l from 0 to b-1 do
        printf("%d,",coeff(gx,x,l)) ;
    end do:
    printf("\n") ;
end do:

Mathematica

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]]]];
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 *)