cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A379430 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces, n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 14, 23, 14, 3, 6, 42, 108, 108, 42, 6, 14, 140, 501, 761, 501, 140, 14, 34, 473, 2264, 4744, 4744, 2264, 473, 34, 95, 1670, 10087, 27768, 38495, 27768, 10087, 1670, 95, 280, 5969, 44310, 153668, 279698, 279698, 153668, 44310, 5969, 280
Offset: 1

Views

Author

Andrew Howroyd, Jan 13 2025

Keywords

Comments

The planar maps considered are connected and may contain loops and parallel edges.
The number of edges is n + k - 2.

Examples

			Array begins:
=========================================================
n\k |  1    2     3      4      5      6      7     8 ...
----+----------------------------------------------------
  1 |  1    1     1      2      3      6     14    34 ...
  2 |  1    2     5     14     42    140    473  1670 ...
  3 |  1    5    23    108    501   2264  10087 44310 ...
  4 |  2   14   108    761   4744  27768 153668 ...
  5 |  3   42   501   4744  38495 279698 ...
  6 |  6  140  2264  27768 279698 ...
  7 | 14  473 10087 153668 ...
  8 | 34 1670 44310 ...
   ...
As a triangle, rows give the number of edges (first row is 0 edges):
   1;
   1,    1;
   1,    2,     1;
   2,    5,     5,     2;
   3,   14,    23,    14,     3;
   6,   42,   108,   108,    42,     6;
  14,  140,   501,   761,   501,   140,    14;
  34,  473,  2264,  4744,  4744,  2264,   473,   34;
  95, 1670, 10087, 27768, 38495, 27768, 10087, 1670, 95;
  ...

Links

Crossrefs

Antidiagonal sums are A006384.
Columns 1..2 are A002995, A380237.
Cf. A269920 (rooted), A277741 (unsensed), A379431 (achiral), A342061 (2-connected), A384964 (simple).

Formula

A(n,k) = A(k,n).

A380239 Number of unsensed planar maps with n vertices and 2 faces.

Original entry on oeis.org

1, 2, 5, 13, 35, 104, 315, 1021, 3407, 11814, 41893, 151688, 556432, 2063446, 7709381, 28977788, 109421539, 414759097, 1577080457, 6013019088, 22980514005, 88012484058, 337717418145, 1298113689274, 4997561829650, 19267942661664, 74386901833067, 287540841925770
Offset: 1

Views

Author

Andrew Howroyd, Jan 19 2025

Keywords

Comments

Also by duality the number of unsensed planar maps with n faces and 2 vertices.
The number of edges is n.

Links

Crossrefs

Column 2 of A277741.
Cf. A380237 (sensed), A380238 (achiral).

Programs

  • PARI
    G1(n)={my(g=(1-sqrt(1-4*x^2 + O(x*x^n)))/(2*x^2)); ((1 + x/(1-x-x^2*g)^2)^2/(1 - x^2*g^2) - 1)/2 + 1/(1 - x*g) - 1}
G2(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); sum(k=1, n, my(m=1+k%2); -(log(2 - c(k)) + log(1 - x^k*c(m*k)^(2/m)))*eulerphi(k)/k, O(x*x^n))}
seq(n)={Vec(G1(n)+G2(n))/4}

Formula

a(n) = (A380237(n) + A380238(n))/2.

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.

Original entry on oeis.org

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

Views

Author

Andrew Howroyd, Jan 28 2025

Keywords

Comments

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

Examples

			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;
  ...

Links

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).

Programs

  • 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])) }

A380618 Number of sensed combinatorial maps with n edges and 2 vertices.

Original entry on oeis.org

1, 2, 8, 38, 275, 2614, 30346, 415360, 6513999, 115063118, 2259975228, 48860184539, 1153140907207, 29502289676802, 813371784160602, 24040797257734161, 758379326971459945, 25432414455826532993, 903508909333199982128, 33897272145242834426910, 1339265974992611047296679
Offset: 1

Views

Author

Andrew Howroyd, Jan 28 2025

Keywords

Comments

By duality, also the number of sensed combinatorial maps with n edges and 2 faces.

Links

Crossrefs

Column 2 of A380615.
Cf. A380237 (planar), A380619 (3 vertices), A380620 (unsensed).

Programs

  • PARI
    \\ Needs G(), InvEulerMTS from A380615.
seq(n, k=2)={my(y='y); Vec(polcoef(InvEulerMTS(G(n, y*(1 + O(y^k)))), k, y))}

A380238 Number of achiral planar maps with n vertices and 2 faces.

Original entry on oeis.org

1, 2, 5, 12, 28, 68, 157, 372, 845, 1949, 4367, 9880, 21858, 48679, 106612, 234546, 509246, 1109352, 2391299, 5167423, 11070598, 23762557, 50641725, 108085708, 229303142, 487039228, 1029167119, 2176808877, 4583856878, 9660020146, 20279242545, 42599286814
Offset: 1

Views

Author

Andrew Howroyd, Jan 19 2025

Keywords

Comments

Also by duality the number of achiral planar maps with n faces and 2 vertices.
The number of edges is n.

Links

Crossrefs

Column 2 of A379431.
Cf. A380237 (sensed), A380239 (unsensed).

Programs

  • PARI
    G1(n)={my(g=(1-sqrt(1-4*x^2 + O(x*x^n)))/(2*x^2)); ((1 + x/(1-x-x^2*g)^2)^2/(1 - x^2*g^2) - 1)/2}
G2(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); sum(k=1, n, my(m=1+k%2); -log(1 - x^k*c(m*k)^(2/m))*eulerphi(k)/k, O(x*x^n))}
seq(n)={Vec(G1(n)+G2(n))/2}

A384966 Number of sensed simple planar maps with n vertices and 2 faces.

Original entry on oeis.org

0, 0, 1, 2, 8, 29, 113, 444, 1763, 6951, 27395, 107672, 422330, 1654180, 6472518, 25308760, 98923442, 386589398, 1510737079, 5904291401, 23079308104, 90236258057, 352908128341, 1380632536468, 5403055984114, 21152009997924, 82835786189975, 324518950873991, 1271797441923614, 4985982054721119
Offset: 1

Views

Author

Andrew Howroyd, Jun 14 2025

Keywords

Comments

In other words, a(n) is the number of embeddings on the sphere of connected simple unicyclic planar graphs with n nodes up to orientation preserving isomorphisms.

Links

Crossrefs

Column 2 of A384964.
Cf. A001429, A006078 (cycle is loop), A007595 (cycle is digon), A380237 (not necessarily simple), A384967 (unsensed version)..

Programs

  • PARI
    seq(n)={my(c(d)=(1-sqrt(1-4*x^d + O(x*x^(n+d))))/(2*x^d)); Vec(1/(1 - x*c(2)) - x*(c(1)^2 + c(2)) - x^2*(c(1)^4 + 3*c(2)^2)/2 - 1 - sum(k=1, n, log(2 - c(k))*eulerphi(k)/k), -n)/2}

Formula

a(n) = A380237(n) - A007595(n) - A006078(n).
Showing 1-6 of 6 results.

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