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Search: id:A033504
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| A033504 |
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a(n)/4^n is expected number of tosses of a coin required to obtain n heads or n tails. |
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+0
4
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| 1, 10, 66, 372, 1930, 9516, 45332, 210664, 960858, 4319100, 19188796, 84438360, 368603716, 1598231992, 6889682280, 29551095248, 126193235194, 536799072924, 2275560109868, 9616650989560, 40527780684972, 170368957887656
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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The number of rooted two-vertex n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets (liskov(AT)im.bas-net.by), Mar 17 2005
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REFERENCES
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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.
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LINKS
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V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.
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FORMULA
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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) = 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 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 (benoit7848c(AT)orange.fr), Mar 05 2005
a(n) = (n+1)*(2^(2*n+1)-binomial(2*n+1,n+1)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 23 2007
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CROSSREFS
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Cf. A002457, A100511, A103943.
Cf. A000346, A130783.
Sequence in context: A080421 A004310 A026853 this_sequence A163615 A117305 A108275
Adjacent sequences: A033501 A033502 A033503 this_sequence A033505 A033506 A033507
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KEYWORD
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easy,nonn,nice
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AUTHOR
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Michael Ulm (ulm(AT)mathematik.uni-ulm.de)
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