A380615 Triangle read by rows: T(n,k) is the number of sensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

1, 1, 1, 2, 2, 1, 5, 8, 5, 2, 18, 38, 34, 14, 3, 105, 275, 288, 154, 42, 6, 902, 2614, 3102, 1959, 705, 140, 14, 9749, 30346, 39242, 27898, 11956, 3142, 473, 34, 127072, 415360, 573654, 446078, 217000, 68544, 13886, 1670, 95, 1915951, 6513999, 9484003, 7911844, 4230802, 1523176, 373188, 60614, 5969, 280

Offset

0,4

Comments

By duality, also the number of sensed combinatorial maps with n edges and k faces.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Example

Triangle begins:
n\k |      1       2       3       4       5      6      7     8   9
----+----------------------------------------------------------------
  0 |      1
  1 |      1,      1
  2 |      2,      2,      1;
  3 |      5,      8,      5,      2;
  4 |     18,     38,     34,     14,      3;
  5 |    105,    275,    288,    154,     42,     6;
  6 |    902,   2614,   3102,   1959,    705,   140,    14;
  7 |   9749,  30346,  39242,  27898,  11956,  3142,   473,   34;
  8 | 127072, 415360, 573654, 446078, 217000, 68544, 13886, 1670, 95;
  ...

Prog

(PARI)
InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
b(k, r)={if(k%2, if(r%2, 0, my(j=r/2); k^j*(2*j)!/(j!*2^j)), sum(j=0, r\2, binomial(r, 2*j)*k^j*(2*j)!/(j!*2^j)))}
C(k, r, y)={my(p=sumdiv(k, d, eulerphi(k/d)*y^d)/k); sum(i=0, r, abs(stirling(r, i, 1))*p^i)/r!}
S(n, k, y)={sum(r=0, 2*n\k, if(k*r%2==0, x^(k*r/2)*b(k, r)*C(k, r, y)), O(x*x^n))}
G(n, y='y)={prod(k=1, 2*n, S(n, k, y))}
T(n)={[Vecrev(p/y) | p<-Vec(y+InvEulerMTS(G(n)))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) }

Crossrefs

Row sums are A170946.

Main diagonal is A002995(n+1).

Second diagonal gives A380237.

Columns 1..3 are A007769, A380618, A380619.

Cf. A053979 (rooted), A379430 (planar), A380616 (unsensed), A380617 (achiral).

Sequence in context: A344528 A380617 A380616 * A329429 A387609 A326617

Adjacent sequences: A380612 A380613 A380614 * A380616 A380617 A380618

Keyword

nonn,tabl

Author

Andrew Howroyd, Jan 28 2025

Status

approved