A046650 Number of rooted planar maps.

1, 1, 2, 4, 14, 49, 216, 984, 4862, 24739, 130338, 701584, 3852744, 21489836, 121525520, 695307888, 4019381790, 23446201495, 137875564710, 816646459860, 4868578092510, 29196022525905, 176022392938080, 1066433501134560, 6490009570139784, 39659537885087124, 243278423033093336, 1497584057249141728, 9249144367260811824

Offset

2,3

Comments

From R. J. Mathar, Apr 13 2019: (Start)

Table III with row sums A000087 is (A046653 row-reversed):

1;

1, 1;

2, 1, 1;

4, 3, 2, 1;

14, 12, 8, 2, 1;

49, 43, 30, 12, 3, 1;

216, 189, 134, 63, 22, 3, 1;

984, 888, 608, 323, 133, 31, 4, 1;

4862, 4332, 2988, 1671, 759, 238, 48, 4, 1;

...

(End)

Links

Table of n, a(n) for n=2..30.

W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.

Formula

Reference gives generating functions.

Maple

B1nm := proc(n, m) # eq (4.15)
    local j ;
    if m>=2 and n>= m  then
        add((3*m-2*j-1)*(2*j-m)*(j-2)!*(3*n-j-m-1)!/(n-j)!/(j-m)!/(j-m+1)!/(2*m-j)!, j=m..min(n, 2*m) ) ;
        %*m/(2*n-m)! ;
    else
        0 ;
    end if;
end proc:
B2wj := proc(w, j) # eq (8.21)
    local k ;
    if  w >= j and j>=1 and w >= 1 then
        add((2*k-j+1)*(k-1)!*(3*w-k-j)!/(k-j+1)!/(k-j)!/(2*j-k-1)!/(w-k)!, k=j..min(w, 2*j-1) ) ;
        %*j/(2*w-j+1)! ;
    else
        0;
    end if;
end proc:
Brwj := proc(r, w, j) # eq. (8.21)
    local k ;
    if  w >= j and j>=1 and w>=1 and r > 1 then
        add((2*k-j)*(k-1)!*(3*w-k-j-1)!/((k-j)!)^2/(2*j-k)!/(w-k)!, k=j..min(w, 2*j) ) ;
        %*j/(2*w-j)! ;
    else
        0 ;
    end if;
end proc:
Brnm := proc(r, n, m)
    if r = 1 then
        B1nm(n, m) ;
    elif r = 2 and type(n, 'odd') and type (m, 'even') then
        B2wj((n-1)/2, m/2) ;
    elif modp(n, r) <> 0 or modp(m, r) <> 0 then
        0;
    else
        Brwj(r, n/r, m/r) ;
    end if;
end proc:
L := proc(n, m) # eq. (6.7)
    add(numtheory[phi](s)*Brnm(s, n, m), s=numtheory[divisors](m)) ;
    %/m ;
end proc:
seq(L(n, 2), n=2..40) ; # R. J. Mathar, Apr 13 2019

Mathematica

B1nm[n_, m_] := If[m >= 2 && n >= m, Sum[(3m - 2j - 1)(2j - m)(j - 2)! (3n - j - m - 1)!/(n - j)!/(j - m)!/(j - m + 1)!/(2m - j)!, {j, m, Min[n, 2m] }] m/(2n - m)!, 0];
B2wj[w_, j_]  := If[w >= j && j >= 1 && w >= 1, Sum[(2k - j + 1)(k - 1)! (3 w - k - j)!/(k - j + 1)!/(k - j)!/(2j - k - 1)!/(w - k)!, {k, j, Min[w, 2 j - 1] }] j/(2w - j + 1)!, 0];
Brwj[r_, w_, j_] := If[w >= j && j >= 1 && w >= 1 && r > 1 , Sum[(2k - j)(k - 1)! (3w - k - j - 1)!/((k - j)!)^2/(2j - k)!/(w - k)!, {k, j, Min[w, 2j]}] j/(2w - j)!, 0];
Brnm[r_, n_, m_] := Which[r == 1, B1nm[n, m], r == 2 && OddQ[n] && EvenQ[m], B2wj[(n - 1)/2, m/2], Mod[n, r] != 0 || Mod[m, r] != 0, 0, True, Brwj[r, n/r, m/r]];
L[n_, m_] := Sum[EulerPhi[s] Brnm[s, n, m], {s, Divisors[m]}]/m;
Table[L[n, 2], {n, 2, 30}] // Flatten (* Jean-François Alcover, Apr 05 2020, after R. J. Mathar *)

Crossrefs

Cf. A000087.

Sequence in context: A032222 A212268 A092665 * A327235 A055727 A003500

Adjacent sequences: A046647 A046648 A046649 * A046651 A046652 A046653

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from R. J. Mathar, Apr 13 2019

Status

approved