cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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.

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%I A104184 #59 Mar 29 2024 08:46:10
%S A104184 1,1,3,9,32,120,473,1925,8034,34188,147787,647141,2864508,12796238,
%T A104184 57615322,261197436,1191268350,5462080688,25162978925,116414836445,
%U A104184 540648963645,2519574506595,11779011525030,55225888341334,259612579655392,1223396051745310
%N 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.
%C A104184 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.
%C A104184 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
%H A104184 Alois P. Heinz, <a href="/A104184/b104184.txt">Table of n, a(n) for n = 0..1437</a>
%H A104184 C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, <a href="https://arxiv.org/abs/1609.06473">Explicit formulas for enumeration of lattice paths: basketball and the kernel method</a>, arXiv preprint arXiv:1609.06473 [math.CO], 2016.
%H A104184 AJ Bu and Doron Zeilberger, <a href="https://arxiv.org/abs/2305.09030">Using Symbolic Computation to Explore Generalized Dyck Paths and Their Areas</a>, arXiv:2305.09030 [math.CO], 2023.
%H A104184 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 512
%H A104184 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.1006/jcta.2002.3273">The tennis ball problem</a>, J. Combin. Theory, A 99 (2002), pp. 307-344.
%F A104184 G.f. (for offset 1): (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))))). - _N-E. Fahssi_, Jan 10 2008
%F A104184 Let M be the infinite pentadiagonal matrix with all 1's in the 1st and 2nd subdiagonals, the main diagonal, and the 1st and 2nd superdiagonals, and with the rest 0's. V = vector [1,0,0,0,...]. The sequence starting with offset 1 = iterates of M*V, leftmost column. - _Gary W. Adamson_, Jun 06 2011
%F A104184 From _Paul D. Hanna_, Oct 19 2011: (Start)
%F A104184 Logarithmic derivative yields the central pentanomial coefficients (A005191).
%F A104184 G.f.: exp( Sum_{n>=1} A005191(n)*x^n/n ).
%F A104184 G.f.: (1/x)*Series_Reversion(x*(1-x^5)*(1-x^2)*(1-x)/(1-x^10)).
%F A104184 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)
%F A104184 a(n) ~ (3-sqrt(5)) * 5^(n+1) / (4 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 09 2014
%F A104184 a(n) = ((Sum_{l=1..n+1} (C(n+1,l)*Sum_{i=0..n-1} (C(i,2*l-1) * Sum_{j=0..n-l+1} (C(j,n-j-i-1)*C(n-l+1,j))))) + Sum_{j=0..n+1} (C(j,n-j) * C(n+1,j)))/(n+1). - _Vladimir Kruchinin_, Jun 26 2015
%F A104184 -2*(n-1)*(2*n+1)*(n+2)*(n+1)*a(n) +(n+1)*(43*n^3-48*n^2-7*n+2)*a(n-1) +(-124*n^4+370*n^3-255*n^2-15*n+14)*a(n-2) +5*(n-2)*(2*n^3-52*n^2+65*n-1)*a(n-3) +25*(n-2)*(n-3)*(8*n^2-8*n-1)*a(n-4) -125*n*(n-2)*(n-3)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Jul 23 2017
%e A104184 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.
%p A104184 a:= proc(n) option remember; `if`(n<5, [1, 1, 3, 9, 32][n+1],
%p A104184       ((n+1)*(43*n^3-48*n^2-7*n+2)*a(n-1)
%p A104184       -(124*n^4-370*n^3+255*n^2+15*n-14)*a(n-2)
%p A104184       +5*(n-2)*(2*n^3-52*n^2+65*n-1)*a(n-3)
%p A104184       +25*(n-2)*(n-3)*(8*n^2-8*n-1)*a(n-4)
%p A104184       -125*n*(n-2)*(n-3)*(n-4)*a(n-5))/
%p A104184        (2*(n-1)*(n+1)*(n+2)*(2*n+1)))
%p A104184     end:
%p A104184 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 11 2013
%t A104184 CoefficientList[Series[(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])])/(4*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 09 2014 *)
%o A104184 (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_ */
%o A104184 (Maxima)
%o A104184 a(n):=((sum(binomial(n+1,l)*sum(binomial(i,2*l-1)*sum(binomial(j,n-j-i-1) *binomial(n-l+1,j),j,0,n-l+1),i,0,n-1),l,1,n+1))+sum(binomial(j,n-j) *binomial(n+1,j),j,0,n+1))/(n+1); /* _Vladimir Kruchinin_, Jun 26 2015 */
%Y A104184 Cf. A066357, A001006, A005191.
%K A104184 nonn
%O A104184 0,3
%A A104184 Nicholas Biller (billern(AT)gmail.com), Mar 11 2005