|
Search: id:A033504
|
|
|
| A033504 |
|
a(n)/4^n is expected number of tosses of a coin required to obtain n heads or n tails. |
|
+0
4
|
|
| 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)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
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
|
|
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
|
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) = 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
|
|
CROSSREFS
|
Cf. A002457, A100511, A103943.
Cf. A000346, A130783.
Adjacent sequences: A033501 A033502 A033503 this_sequence A033505 A033506 A033507
Sequence in context: A080421 A004310 A026853 this_sequence A163615 A117305 A108275
|
|
KEYWORD
|
easy,nonn,nice
|
|
AUTHOR
|
Michael Ulm (ulm(AT)mathematik.uni-ulm.de)
|
|
|
Search completed in 0.002 seconds
|
