

A033504


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


5



1, 10, 66, 372, 1930, 9516, 45332, 210664, 960858, 4319100, 19188796, 84438360, 368603716, 1598231992, 6889682280, 29551095248, 126193235194, 536799072924, 2275560109868, 9616650989560, 40527780684972, 170368957887656
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OFFSET

0,2


COMMENTS

The number of rooted twovertex nedge 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. 127129.
V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 200501, 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), 364387.


FORMULA

With a different offset: Sum_{j=0..n} Sum_{k=0..n} binomial(n, j)*binomial(n, k)*min(j, k) = n*2^(n1) + (n/2)*binomial(2*n, n) [see Klamkin]
a(n1) = b(n, n), where b(n, m) = b(n1, m)/2+b(n, m1)/2+1; b(n, 0)=b(0, n)=0
a(n) = sum 2^(2 n  k  l) Binomial(k+l, k), where the sum is from 0 to n for k and l
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(n1) +48*(n1)*a(n2) +32*(2*n+3)*a(n3)=0.  R. J. Mathar, Dec 22 2013


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.uniulm.de)


STATUS

approved



