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

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

0,2

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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..172

V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

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

Example

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)

Mathematica

a[n_]:=(n+1)*(2^(2*n+1)-Binomial[2*n+1, n+1])
a /@ Range[0, 50] (* Julien Kluge, Jul 21 2016 *)

Prog

(Magma) [(n+1)*(2^(2*n+1)-Binomial(2*n+1, n+1)): n in [0..25]]; // Vincenzo Librandi, Jun 09 2011

Crossrefs

Cf. A002457, A100511, A103943.

Cf. A000346, A130783.

Sequence in context: A004310 A026853 A177452 * A163615 A232062 A229003

Adjacent sequences: A033501 A033502 A033503 * A033505 A033506 A033507

Keyword

easy,nonn,nice

Author

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

Extensions

Name corrected by Jeremy Tan, Mar 13 2018

Status

approved