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A198258 Row sums of A197655.

%I A198258 #56 Apr 17 2023 11:08:55
%S A198258 1,7,190,5831,219626,8976562,394800204,18211045575,873216720082,
%T A198258 43136486269382,2183722698469676,112795257850321202,
%U A198258 5925951358743662260,315869014732813229716,17048301919464100932440,930217336628892162216135,51244644190683748419791970
%N A198258 Row sums of A197655.
%C A198258 Number of meanders of length (n+1)*6 which are composed by arcs of equal length and a central angle of 60 degrees.
%C A198258 Definition of a meander:
%C A198258 A binary curve C is a triple (m, S, dir) such that
%C A198258 (a) S is a list with values in {L,R} which starts with an L,
%C A198258 (b) dir is a list of m different values, each value of S being allocated a value of dir,
%C A198258 (c) consecutive Ls increment the index of dir,
%C A198258 (d) consecutive Rs decrement the index of dir,
%C A198258 (e) the integer m>0 divides the length of S and
%C A198258 (f) C is a meander if each value of dir occurs length(S)/m times.
%C A198258 For this sequence, m = 6.
%C A198258 The terms are proved by brute force for 0 <= n <= 5, but not yet in general. - _Susanne Wienand_, Oct 29 2011
%H A198258 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/Meander">Meanders and walks on the circle</a>.
%F A198258 a(n) = Sum_{k=0..n} Sum_{j=0..5} Sum_{i=0..5} (-1)^(j+i)*C(i,j)*C(n,k)^6*(n+1)^j*(k+1)^(5-j)/(k+1)^5. - _Peter Luschny_, Nov 02 2011
%F A198258 a(n) = Sum_{k=0..n} h(n,k)*binomial(n,k)^6, where h(n,k) = (1+k)*(1-((n-k)/(1+k))^6)/(1+2*k-n) if 1+2*k-n <> 0 else h(n,k) = 6. - _Peter Luschny_, Nov 24 2011
%F A198258 Conjecture: working with offset 1, that is, a(1) = 1, a(2) = 7, ..., then the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for positive integers n and r and all primes p >= 3. - _Peter Bala_, Mar 21 2023
%F A198258 a(n) ~ sqrt(3) * 2^(6*n+3) / (Pi*n)^(5/2). - _Vaclav Kotesovec_, Apr 17 2023
%e A198258 Some examples of list S and allocated values of dir if n = 4:
%e A198258 Length(S) = (4+1)*6 = 30.
%e A198258   S: L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L
%e A198258 dir: 1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,0
%e A198258   S: L,L,L,L,L,L,L,R,R,R,R,L,R,R,R,R,R,L,L,L,L,R,R,L,L,L,L,R,L,L
%e A198258 dir: 1,2,3,4,5,0,1,1,0,5,4,4,4,3,2,1,0,0,1,2,3,3,2,2,3,4,5,5,5,0
%e A198258   S: L,L,L,L,L,R,R,R,R,R,R,R,L,L,R,L,L,R,L,L,R,L,L,L,R,R,L,L,L,L
%e A198258 dir: 1,2,3,4,5,5,4,3,2,1,0,5,5,0,0,0,1,1,1,2,2,2,3,4,4,3,3,4,5,0
%e A198258 Each value of dir occurs 30/6 = 5 times.
%p A198258 A198258 := proc(n) local i, j, k, pow;
%p A198258 pow := (a, b) -> if a=0 and b=0 then 1 else a^b fi;
%p A198258 add(add(add((-1)^(j+i)*binomial(i,j)*binomial(n,k)^6*pow(n+1,j)*pow(k+1,5-j)/(k+1)^5, i=0..5),j=0..5),k=0..n) end:
%p A198258 seq(A198258(n), n=0..16); # _Peter Luschny_, Nov 02 2011
%t A198258 T[n_, k_] := (1 + n)(1 + 3 k + 3 k^2 - n - 3 k*n + n^2)(1 + k + k^2 + n - k*n + n^2) Binomial[n, k]^6/(1 + k)^5;
%t A198258 a[n_] := Sum[T[n, k], {k, 0, n}];
%t A198258 Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Jun 29 2019 *)
%o A198258 (PARI)
%o A198258 A198258(n) = {sum(k=0,n,if(n == 1+2*k,6,(1+k)*(1-((n-k)/(1+k))^6)/(1+2*k-n))*binomial(n,k)^6)} \\ _Peter Luschny_, Nov 24 2011
%Y A198258 Cf. A198060, A198256, A198257.
%K A198258 nonn
%O A198258 0,2
%A A198258 _Susanne Wienand_, Oct 24 2011