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 1-4 of 4 results.

A380237 Number of sensed planar maps with n vertices and 2 faces.

Original entry on oeis.org

1, 2, 5, 14, 42, 140, 473, 1670, 5969, 21679, 79419, 293496, 1091006, 4078213, 15312150, 57721030, 218333832, 828408842, 3151769615, 12020870753, 45949957412, 176001205559, 675384194565, 2596119292840, 9994894356158, 38535398284100, 148772774499015, 575079507042663
Offset: 1

Views

Author

Andrew Howroyd, Jan 19 2025

Keywords

Comments

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

Links

Crossrefs

Column 2 of A379430.
Cf. A000346 (rooted), A380238 (achiral), A380239 (unsensed), A060404 (with a distinguished face), A103943 (with a distinguished vertex).
Cf. A000108, A210736.

Programs

  • PARI
    a(n) = {(binomial(n - 1, (n - 1)\2) + sumdiv(n, d, eulerphi(n/d)*(2^(2*d-1) - binomial(2*d-1, d)))/n)/2}
  • 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)) - 1 - sum(k=1, n, log(2 - c(k))*eulerphi(k)/k))/2}

Formula

a(n) = (A210736(n) + A060404(n))/2.
a(n) = (1/(2*n))*(n*binomial(n-1, floor((n-1)/2)) + Sum_{d|n} phi(n/d)*(2^(2*d-1) - binomial(2*d-1, d))).
G.f.: (1/2)*(1/(1 - x*C(x^2)) - 1 - Sum_{k>=1} log(1 - C(x^k)) * phi(k)/k), where C(x) is the g.f. of A000108.

A033504 a(n)/4^n is the expected number of tosses of a coin required to obtain n+1 heads or n+1 tails.

Original entry on oeis.org

1, 10, 66, 372, 1930, 9516, 45332, 210664, 960858, 4319100, 19188796, 84438360, 368603716, 1598231992, 6889682280, 29551095248, 126193235194, 536799072924, 2275560109868, 9616650989560, 40527780684972, 170368957887656, 714556104675736, 2990728476330672
Offset: 0

Views

Author

Michael Ulm (ulm(AT)mathematik.uni-ulm.de)

Keywords

Comments

The number of rooted two-vertex n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005

Examples

			From _Jeremy Tan_, Mar 13 2018: (Start)
For n=1 the sequences of flips ending at two heads or two tails are:
HH, TT (probability 1/4 each)
HTH, HTT, THH, THT (1/8 each)
The expected number of flips is 2*2*1/4 + 3*4*1/8 = 10/4 = a(1)/4^1. (End)

References

  • M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 127-129.
  • V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Links

Crossrefs

Cf. A002457, A100511, A103943.
Cf. A000346, A130783.

Programs

  • Magma
    [(n+1)*(2^(2*n+1)-Binomial(2*n+1,n+1)): n in [0..25]]; // Vincenzo Librandi, Jun 09 2011
  • Mathematica
    a[n_]:=(n+1)*(2^(2*n+1)-Binomial[2*n+1,n+1])
a /@ Range[0,50] (* Julien Kluge, Jul 21 2016 *)

Formula

With a different offset: Sum_{j=0..n} Sum_{k=0..n} binomial(n, j)*binomial(n, k)*min(j, k) = n*2^(n-1) + (n/2)*binomial(2*n, n). [see Klamkin]
a(n-1) = 4^(n-1)*b(n, n), where b(n, m) = b(n-1, m)/2 + b(n, m-1)/2 + 1; b(n, 0)=b(0, n)=0.
a(n) = Sum_{k=0..n, l=0..n} 2^(2n - k - l) binomial(k+l, k).
a(n) = (2n+1)*Sum_{0<=i,j<=n} binomial(2n, i+j)/(i+j+1). - Benoit Cloitre, Mar 05 2005
a(n) = (n+1)*(2^(2*n+1) - binomial(2*n+1,n+1)). - Vladeta Jovovic, Aug 23 2007
n*a(n) + 6*(-2*n+1)*a(n-1) + 48*(n-1)*a(n-2) + 32*(-2*n+3)*a(n-3) = 0. - R. J. Mathar, Dec 22 2013
a(n) ~ 2^(2*n+1)*n. - Ilya Gutkovskiy, Jul 21 2016

Extensions

Name corrected by Jeremy Tan, Mar 13 2018

A060404 G.f.: Sum_{k >= 1} (phi(k)/k)*log(1-f(x^k)), where f(x) = (1 - sqrt(1 - 4*x)) / (2*x) - 1 is the g.f. for the Catalan numbers (A000108) C_1, C_2, C_3, ...

Original entry on oeis.org

0, 1, 3, 8, 25, 78, 270, 926, 3305, 11868, 43232, 158586, 586530, 2181088, 8154710, 30620868, 115435625, 436654794, 1656793374, 6303490610, 24041649128, 91899730068, 352002058402, 1350767683698, 5192237233602, 19989786008160
Offset: 0

Views

Author

N. J. A. Sloane, Apr 05 2001

Keywords

Comments

Counts cycles of objects where the individual objects are anything enumerated by the Catalan numbers C_1, C_2, ...
The number of unrooted two-face n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005

References

  • V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Links

Crossrefs

Cf. A103943.

Programs

  • Mathematica
    max = 25; f[x_] := (1 - Sqrt[1 - 4*x])/(2*x) - 1; gf = Sum[(EulerPhi[k]/k)*Log[1 - f[x^k]], {k, 1, max}]; CoefficientList[ Series[-gf, {x, 0, max}], x] (* Jean-François Alcover, Jan 21 2013 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*(2^(2*d-1) - binomial(2*d-1, d)))/n; \\ Andrew Howroyd, Apr 02 2017

Formula

a(n) = (1/n) * Sum_{d|n} phi(n/d) * A000346(d-1) for n>0. - Andrew Howroyd, Apr 02 2017

A380240 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces including one distinguished outside face, n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 12, 8, 2, 10, 48, 64, 25, 3, 26, 196, 412, 314, 78, 6, 80, 798, 2458, 2976, 1478, 270, 14, 246, 3248, 13452, 23588, 18844, 6748, 926, 34, 810, 13184, 70330, 166050, 192096, 110714, 30168, 3305, 95, 2704, 53416, 353716, 1074472, 1676668, 1397484, 613884, 132734, 11868, 280, 9252
Offset: 1

Views

Author

Andrew Howroyd, Jan 21 2025

Keywords

Comments

The number of edges is n + k - 2.

Examples

			Array begins:
==============================================================
n\k |  1    2      3      4       5       6       7      8 ...
----+---------------------------------------------------------
  1 |  1    1      2      4      10      26      80    246 ...
  2 |  1    3     12     48     196     798    3248  13184 ...
  3 |  1    8     64    412    2458   13452   70330 353716 ...
  4 |  2   25    314   2976   23588  166050 1074472 ...
  5 |  3   78   1478  18844  192096 1676668 ...
  6 |  6  270   6748 110714 1397484 ...
  7 | 14  926  30168 613884 ...
  8 | 34 3305 132734 ...
   ...

Crossrefs

Columns 1..2 are A002995, A060404.
Rows 1..2 are A003239(n-1), A103943.
Antidiagonal sums are A103937.
Cf. A269920 (rooted), A379430 (sensed with no root).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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