cp's OEIS Frontend

A197657 Row sums of A194595.

%I A197657 #95 Apr 23 2023 12:19:41
%S A197657 1,4,22,134,866,5812,40048,281374,2006698,14482064,105527060,
%T A197657 775113440,5731756720,42628923040,318621793472,2391808860446,
%U A197657 18023208400634,136271601087352,1033449449559724,7858699302115444,59906766929537116,457685157123172664
%N A197657 Row sums of A194595.
%C A197657 Number of meanders of length (n+1)*3 which are composed by arcs of equal length and a central angle of 120 degrees.
%C A197657 Definition of a meander:
%C A197657 A binary curve C is a triple (m, S, dir) such that
%C A197657 (a) S is a list with values in {L,R} which starts with an L,
%C A197657 (b) dir is a list of m different values, each value of S being allocated a value of dir,
%C A197657 (c) consecutive Ls increment the index of dir,
%C A197657 (d) consecutive Rs decrement the index of dir,
%C A197657 (e) the integer m>0 divides the length of S and
%C A197657 (f) C is a meander if each value of dir occurs length(S)/m times.
%C A197657 For this sequence, m = 3.
%C A197657 For 0 <= n <= 16, a(n) = the hypergraph Fuss-Catalan number FC_1^(2,n+1) in the notation of Chavan et al. - see 7.1 in the Appendix. - _Peter Bala_, Apr 11 2023
%H A197657 Parth Chavan, Andrew Lee and Karthik Seetharaman, <a href="https://arxiv.org/abs/2202.01111">Hypergraph Fuss-Catalan numbers</a>, arXiv:2202.01111 [math.CO]
%H A197657 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/Meander">Meanders and walks on the circle</a>.
%H A197657 Susanne Wienand, <a href="http://oeis.org/wiki/File:Meander_m%3D3.gif">Animation of a meander</a>
%H A197657 Susanne Wienand, <a href="https://oeis.org/wiki/File:Meander,_m%3D3.png">Example of a meander</a>
%F A197657 a(n) = Sum{k=0..n} Sum{j=0..2} Sum{i=0..2} (-1)^(j+i)*C(i,j)*C(n,k)^3*(n+1)^j*(k+1)^(2-j)/(k+1)^2. - _Peter Luschny_, Nov 02 2011
%F A197657 a(n) = Sum_{k=0..n} h(n,k)*binomial(n,k)^3, where h(n,k) = (1+k)*(1-((n-k)/(1+k))^3)/(1+2*k-n) if 1+2*k-n <> 0 else h(n,k) = 3. - _Peter Luschny_, Nov 24 2011
%F A197657 a(n) = A141147(n+1)/2 = A110707(n+1)/6 = (A000172(n)+A000172(n+1))/3. - _Max Alekseyev_, Jul 15 2014
%F A197657 Conjecture: (n+1)^2*a(n) -3*(n+1)*(2*n+1)*a(n-1) -3*n*(5*n-7)*a(n-2) -8*(n-2)^2*a(n-3)=0. - _R. J. Mathar_, Jul 26 2014
%F A197657 a(n) = 2^n*hypergeom([n + 1, -n/2, -n/2 - 1/2], [1, 1], 1). - _Peter Luschny_, Mar 26 2023
%F A197657 a(n) ~ sqrt(3) * 2^(3*n+1) / (Pi*n). - _Vaclav Kotesovec_, Apr 17 2023
%e A197657 Some examples of list S and allocated values of dir if n = 4:
%e A197657 Length(S) = (4+1)*3 = 15.
%e A197657   S: L,L,L,L,L,L,L,L,L,L,L,L,L,L,L
%e A197657 dir: 1,2,0,1,2,0,1,2,0,1,2,0,1,2,0
%e A197657   S: L,L,L,L,R,L,L,R,L,L,R,L,L,L,L
%e A197657 dir: 1,2,0,1,1,1,2,2,2,0,0,0,1,2,0
%e A197657   S: L,R,L,L,L,L,L,R,L,L,R,L,R,R,R
%e A197657 dir: 1,1,1,2,0,1,2,2,2,0,0,0,0,2,1
%e A197657 Each value of dir occurs 15/3 = 5 times.
%p A197657 A197657 := proc(n)
%p A197657     (A000172(n) + A000172(n+1)) / 3 ;
%p A197657 end proc; # _R. J. Mathar_, Jul 26 2014
%p A197657 a := n -> 2^n*hypergeom([n + 1, -n/2, -n/2 - 1/2], [1, 1], 1):
%p A197657 seq(simplify(a(n)), n = 0..21); # _Peter Luschny_, Mar 26 2023
%t A197657 A197657[n_] := Sum[Sum[Sum[(-1)^(j + i)* Binomial[i, j]*Binomial[n, k]^3*(n + 1)^j*(k + 1)^(2 - j)/(k + 1)^2, {i, 0, 2}], {j, 0, 2}], {k, 0, n}]; Table[A197657[n], {n, 0, 16}]  (* _Peter Luschny_, Nov 02 2011 *)
%o A197657 (SageMath)
%o A197657 def A197657(n):
%o A197657     return 2^n*hypergeometric([n + 1, -n/2, -n/2 - 1/2], [1, 1], 1).simplify_hypergeometric()
%o A197657 for n in (0..21): print(A197657(n)) # _Peter Luschny_, Mar 26 2023
%o A197657 (PARI)
%o A197657 A197657(n) = {sum(k=0,n,if(n == 1+2*k,3,(1+k)*(1-((n-k)/(1+k))^3)/(1+2*k-n))*binomial(n,k)^3)} \\ _Peter Luschny_, Nov 24 2011
%Y A197657 Cf. A198060, A198256, A198257, A198258.
%K A197657 nonn,easy
%O A197657 0,2
%A A197657 _Susanne Wienand_, Oct 17 2011