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 A197653 Triangle by rows T(n,k), showing the number of meanders with length (n+1)*4 and containing (k+1)*4 Ls and (n-k)*4 Rs, where Ls and Rs denote arcs of equal length and a central angle of 90 degrees which are positively or negatively oriented. 8
 1, 4, 1, 15, 30, 1, 40, 324, 120, 1, 85, 2080, 3120, 340, 1, 156, 9375, 40000, 18750, 780, 1, 259, 32886, 328125, 437500, 82215, 1554, 1, 400, 96040, 1959216, 6002500, 3265360, 288120, 2800, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Definition of a meander: A binary curve C is a triple (m, S, dir) such that (a) S is a list with values in {L,R} which starts with an L, (b) dir is a list of m different values, each value of S being allocated a value of dir, (c) consecutive Ls increment the index of dir, (d) consecutive Rs decrement the index of dir, (e) the integer m > 0 divides the length of S and (f) C is a meander if each value of dir occurs length(S)/m times. For this sequence, m = 4. The values in the triangle are proved by brute force for 0 <= n <= 8. The formulas are not yet proved in general. The number triangle can be calculated recursively by the number triangles A007318, A103371 and A194595. The first column of the triangle seems to be A053698. The diagonal right hand is A000012. The diagonal with k = n-1 seems to be A027445. Row sums are in A198256. The conjectured formulas are confirmed by dynamic programming for 0 <= n <= 43. - Susanne Wienand, Jun 29 2015 LINKS Susanne Wienand, Table of n, a(n) for n = 0..989 Peter Luschny, Meanders and walks on the circle. FORMULA Recursive formula (conjectured): T(n,k)   = T(4,n,k) T(4,n,k) = T(1,n,k)^4 + T(1,n,k)*T(3,n,n-1-k), 0 <= k < n T(4,n,n) = 1                                        k = n T(3,n,k) = T(1,n,k)^3 + T(1,n,k)*T(2,n,n-1-k), 0 <= k < n T(3,n,n) = 1                                        k = n T(2,n,k) = T(1,n,k)^2 + T(1,n,k)*T(1,n,n-1-k), 0 <= k < n T(2,n,n) = 1                                        k = n T(3,n,k) = A194595 T(2,n,k) = A103371 T(1,n,k) = A007318 (Pascal's Triangle) closed formula (conjectured): T(n,n) = 1,                         k = n                 T(n,k) = A + B + C + D,             k < n                      A = (C(n,k))^4                      B = (C(n,k))^3 * C(n,n-1-k)                      C = (C(n,k))^2 *(C(n,n-1-k))^2                      D =  C(n,k)    *(C(n,n-1-k))^3 Let S(n,k) = binomial(2*n,n)^(k+1)*((n+1)^(k+1)-n^(k+1))/(n+1)^k. Then T(2*n,n) = S(n,3). - Peter Luschny, Oct 20 2011 T(n,k) = A198063(n+1,k+1)*C(n,k)^4/(k+1)^3. - Peter Luschny, Oct 29 2011 T(n,k) = h(n,k)*binomial(n,k)^4, where h(n,k) = (1+k)*(1-((n-k)/(1+k))^4)/(1+2*k-n) if 1+2*k-n <> 0, otherwise h(n,k) = 4. - Peter Luschny, Nov 24 2011 EXAMPLE For n = 4 and k = 2, T(4,4,2) = 3120. Recursive example: T(1,4,0) = 1, T(1,4,1) = 4, T(1,4,2) = 6, T(1,4,3) = 4, T(1,4,4) = 1, T(3,4,0) = 21, T(3,4,1) = 304, T(3,4,2) = 456, T(3,4,3) = 84, T(3,4,1) = 1, T(4,4,2) = 6^4 + 6*304 = 3120. Example for closed formula: T(4,2) = 6^4 + 6^3 * 4 + 6^2 * 4^2 + 6 * 4^3 = 3120. Some examples of list S and allocated values of dir if n = 4 and k = 2: Length(S) = (4+1)*4 = 20 and S contains (2+1)*4 = 12 Ls.   S: L,L,L,L,L,L,L,L,L,L,L,L,R,R,R,R,R,R,R,R dir: 1,2,3,0,1,2,3,0,1,2,3,0,0,3,2,1,0,3,2,1   S: L,L,L,R,L,L,R,L,L,R,R,L,L,L,R,L,L,R,R,R dir: 1,2,3,3,3,0,0,0,1,1,0,0,1,2,2,2,3,3,2,1   S: L,R,L,L,L,L,R,R,R,L,L,R,R,L,L,L,R,L,L,R dir: 1,1,1,2,3,0,0,3,2,2,3,3,2,2,3,0,0,0,1,1 Each value of dir occurs 20/4 = 5 times. MAPLE A197653 := (n, k) -> binomial(n, k)^4*(n+1)*(n^2-2*n*k+1+2*k+2*k^2)/((1+k)^3); seq(print(seq(A197653(n, k), k=0..n)), n=0..7); # Peter Luschny, Oct 19 2011 MATHEMATICA T[n_, k_] := Binomial[n, k]^4 (n+1)(n^2 - 2n*k + 1 + 2k + 2k^2)/((1+k)^3); Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018, after Peter Luschny *) PROG (C#) static int[] A197653_row(int r) { return GenBinomial(r, 4); } // The function GenBinomial(r, s) is defined in A194595. // This C# program causes numerical overflow for results larger than 2147483647. - Susanne Wienand, Jun 29 2015 (Sage) def S(N, n, k) : return binomial(n, k)^(N+1)*sum(sum((-1)^(N-j+i)*binomial(N-i, j)*((n+1)/(k+1))^j for i in (0..N) for j in (0..N))) def A197653(n, k) : return S(3, n, k) for n in (0..5) : print [A197653(n, k) for k in (0..n)]  ## Peter Luschny, Oct 24 2011 (PARI) A197653(n, k) = {if(n==1+2*k, 4, (1+k)*(1-((n-k)/(1+k))^4)/(1+2*k-n))*binomial(n, k)^4} \\ Peter Luschny, Nov 24 2011 CROSSREFS Cf. A000012, A007318, A027445, A053698, A103371, A194595, A197654, A197655, A198063, A198256. Sequence in context: A207823 A056920 A123382 * A146160 A059222 A117292 Adjacent sequences:  A197650 A197651 A197652 * A197654 A197655 A197656 KEYWORD nonn,tabl AUTHOR Susanne Wienand, Oct 17 2011 STATUS approved

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Last modified November 19 20:42 EST 2019. Contains 329323 sequences. (Running on oeis4.)