|
Search: id:A005022
|
|
|
| A005022 |
|
Number of walks of length 2n+6 in the path graph P_7 from one end to the other.
(Formerly M4171)
|
|
+0
2
|
|
| 6, 26, 100, 364, 1288, 4488, 15504, 53296, 182688, 625184, 2137408, 7303360, 24946816, 85196928, 290926848, 993379072, 3391793664, 11580678656, 39539651584, 134998297600, 460915984384, 1573671536640, 5372862566400
(list; graph; listen)
|
|
|
OFFSET
|
1,1
|
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Everett, C. J.; Stein, P. R.; The combinatorics of random walk with absorbing barriers. Discrete Math. 17 (1977), no. 1, 27-45.
W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96.
Flajolet, P.; Raoult, J.-C.; Vuillemin, J.; The number of registers required for evaluating arithmetic expressions. Theoret. Comput. Sci. 9 (1979), no. 1, 99-125.
|
|
LINKS
|
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
|
|
FORMULA
|
G.f.: 1/(1-6x+10x^2-4x^3)-1.
a(n)=6a(n-1)-10a(n-2)+4a(n-3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004
a(k)=sum(binomial(6+2k, 8j+k-2)-binomial(6+2k, 8j+k-1), j=-infinity..infinity) (a finite sum).
|
|
EXAMPLE
|
Example: a(1)=6 because in the path ABCDEFG we have ABABCDEFG, ABCBCDEFG, ABCDCDEFG, ABCDEDEFG, ABCDEFEFG and ABCDEFGFG. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004
|
|
MAPLE
|
a:=k->sum(binomial(6+2*k, 8*j+k-2), j=ceil((2-k)/8)..floor((8+k)/8))-sum(binomial(6+2*k, 8*j+k-1), j=ceil((1-k)/8)..floor((7+k)/8)): seq(a(k), k=1..28);
A005022:=-1/((2*z-1)*(2*z**2-4*z+1)) -1; [Conjectured (correctly) by S. Plouffe in his 1992 dissertation. Gives sequence with an additional leading term of 1.]
|
|
CROSSREFS
|
Sequence in context: A055589 A055420 A137746 this_sequence A094811 A125107 A034560
Adjacent sequences: A005019 A005020 A005021 this_sequence A005023 A005024 A005025
|
|
KEYWORD
|
nonn,walk
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
|
|
EXTENSIONS
|
Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 28 2004
|
|
|
Search completed in 0.002 seconds
|
