|
| |
|
|
A104184
|
|
a(n) is the number of paths from (0,0) to (n,0) using steps of the form (1,2),(1,1),(1,0),(1,-1) or (1,-2) and staying above the x-axis. Also, a(n) is the number of possible combinations of balls on the lawn after n turns, using a Motzkin variation of the (4,2)-case of the tennis ball problem considered by D. Merlini, R. Sprugnoli and M. C. Verri.
|
|
3
| |
|
|
1, 1, 3, 9, 32, 120, 473, 1925, 8034, 34188, 147787, 647141, 2864508, 12796238, 57615322, 261197436, 1191268350, 5462080688, 25162978925, 116414836445, 540648963645, 2519574506595, 11779011525030, 55225888341334
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| The (4,2)-case of the Motzkin Tennis Ball Problem is a variation of the Tennis Ball Problem that generates a(n). On each turn, i, four balls labeled i are placed in the bucket and then any two are removed and placed on the lawn. We consider all possible combinations of balls on the lawn after n turns.
The number of ways to choose n numbers, ranging from 0 to 4, so that their sum is 2n and so that when you take k numbers from the left, the sum of these numbers is <= 2k (e.g. the combination of {141} is impossible, for 1+4 > 2k). Thus a(1) = {2}; a(2) = {04}, {13} and {22}; a(3) = {024}, {033}, {042}, {114}, {123}, {132}, {204}, {213} and {222} - Joost Vermeij (joost_vermeij(AT)hotmail.com), Jun 12 2005
|
|
|
REFERENCES
| D. Merlini, R. Sprugnoli, M.C. Verri, The Tennis Ball Problem, Journal of Combinatorial Theory, Series A, Vol. 99 (2002), pp. 307-344.
|
|
|
LINKS
| P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 512
|
|
|
FORMULA
| G.f.: (1/(4*x))*(1+x+sqrt((1-6*x+5*x^2)) - sqrt(2)*sqrt(1+sqrt((1-6*x+5*x^2)) + x*(-2-5*x+sqrt((1-6*x+5*x^2))))). - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Jan 10 2008
Let M = an infinite pentadiagonal matrix with all 1's in the sub-sub, sub, main, super, and super-super diagonals with the rest zeros. V = vector [1,0,0,0,...]. The sequence starting with offset 1 = iterates of M*V, leftmost column. [From Gary W. Adamson, (qntmpkt(AT)yahoo.com), Jun 06 2011]
Contribution from Paul D. Hanna, Oct 19 2011: (Start)
Logarithmic derivative yields the central pentanomial coefficients (A005191).
G.f.: exp( Sum_{n>=1} A005191(n)*x^n/n ).
G.f.: (1/x)*Series_Reversion(x*(1-x^5)*(1-x^2)*(1-x)/(1-x^10)).
G.f. satisfies: A(x) = (1-x^10*A(x)^10)/((1-x^5*A(x)^5)*(1-x^2*A(x)^2)*(1-x*A(x))). (See formula from Michael Somos in A005191) (End)
|
|
|
EXAMPLE
| a(3)=9, since the possible combinations of balls on the lawn after 3 turns is 111122, 111123, 111133, 111222, 111223, 111233, 112222, 112223, 112233, if on each turn there are 4 identically labeled balls received and 2 selected.
|
|
|
PROG
| (PARI) {a(n)=local(A=1); A=exp(sum(m=1, n+1, polcoeff(((1-x^5)/(1-x)+O(x^(2*m+1)))^m, 2*m)*x^m/m)+x*O(x^n)); polcoeff(A, n)} /* Paul D. Hanna */
|
|
|
CROSSREFS
| Cf. A066357, A001006, A005191.
Sequence in context: A192206 A091841 A063020 * A193621 A183425 A039628
Adjacent sequences: A104181 A104182 A104183 * A104185 A104186 A104187
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Nicholas Billler (billern(AT)gmail.com), Mar 11 2005
|
| |
|
|
