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 A194595 Triangle by rows T(n,k), showing the number of meanders with length (n+1)*3 and containing (k+1)*3 Ls and (n-k)*3 Rs, where Ls and Rs denote arcs of equal length and a central angle of 120 degrees which are positively or negatively oriented. 8
 1, 3, 1, 7, 14, 1, 13, 81, 39, 1, 21, 304, 456, 84, 1, 31, 875, 3000, 1750, 155, 1, 43, 2106, 13875, 18500, 5265, 258, 1, 57, 4459, 50421, 128625, 84035, 13377, 399, 1, 73, 8576, 153664, 669536, 836920, 307328, 30016, 584, 1, 91, 15309, 409536, 2815344, 6001128, 4223016, 955584, 61236, 819, 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 = 3. The values in the triangle are proved by brute force for 0 <= n <= 11. The formulas are not yet proved in general. - Susanne Wienand Let S(N,n,k) = C(n,k)^(N+1)*sum_{j=0..N}sum_{i=0..N}(-1)^(N-j+i)*C(N-i,j)*((n+1)/(k+1))^j. Then S(0,n,k) = A007318(n,k), S(1,n,k) = A103371(n,k), S(2,n,k) = T(n,k), S(3,n,k) = A197653(n,k), S(4,n,k) = A197654(n,k), S(5,n,k) = A197655(n,k). - Peter Luschny, Oct 21 2011 The number triangle can be calculated recursively by the number triangles A103371 and A007318. The first column of the triangle contains the central polygonal numbers A002061. The diagonal right hand is A000012. The diagonal with k = n-1 seems to be A027444. Row sums are in A197657. - Susanne Wienand, Nov 24 2011 The conjectured formulas are confirmed by dynamic programming for 0 <= n <= 62. - Susanne Wienand, Jun 24 2015 LINKS Susanne Wienand, Table of n, a(n) for n = 0..2015 Peter Luschny, Meanders and walks on the circle. Susanne Wienand, Animation of a meander Susanne Wienand, Example of a meander FORMULA Recursive formula (conjectured): T(n,k) = 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(2,n,k) = A103371, T(1,n,k) = A007318 (Pascal's Triangle). Closed formula (conjectured): T(n,k) = (C(n,k))^3 + C(n,k) * C(n,k+1) * C(n+1,k+1). [Susanne Wienand] 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,2). [Peter Luschny, Oct 20 2011] T(n,k) = A073254(n+1,k+1)C(n,k)^3/(k+1)^2. [Peter Luschny, Oct 29 2011] T(n,k) = 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] EXAMPLE For n = 4 and k = 2, T(3,4,2) = 456 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(2,4,0) = 5 T(2,4,1) = 40 T(2,4,2) = 60 T(2,4,3) = 20 T(2,4,4) = 1 T(3,4,0) = T(1,4,0)^3 + T(1,4,0)*T(2,4,4-1-0) = 1^3 + 1*20 = 21 T(3,4,1) = T(1,4,1)^3 + T(1,4,1)*T(2,4,4-1-1) = 4^3 + 4*60 = 304 T(3,4,2) = T(1,4,2)^3 + T(1,4,2)*T(2,4,4-1-2) = 6^3 + 6*40 = 456 T(3,4,3) = T(1,4,3)^3 +(T(1,4,3)*T(2,4,4-1-3) = 4^3 + 4*5  = 84 T(3,4,4) = 1 example for closed formula: T(4,2) = (C(4,2))^3 + C(4,2) * C(4,3) * C(5,3) = 6^3 + 6 * 4 * 10 = 456 Some examples of list S and allocated values of dir if n = 4 and k = 2: Length(S) = (4+1)*3 = 15 and S contains (2+1)*3 = 9 Ls.   S: L,L,L,L,L,L,L,L,L,R,R,R,R,R,R dir: 1,2,0,1,2,0,1,2,0,0,2,1,0,2,1   S: L,L,R,L,L,L,L,R,R,L,R,R,L,R,L dir: 1,2,2,2,0,1,2,2,1,1,1,0,0,0,0   S: L,R,R,R,L,L,L,L,R,R,L,L,L,R,L dir: 1,1,0,2,2,0,1,2,2,1,1,2,0,0,0 Each value of dir occurs 15/3 = 5 times. MAPLE A194595 := (n, k)->binomial(n, k)^3*(k^2+k+1+n^2+n-k*n)/((k+1)^2); seq(print(seq(A194595(n, k), k=0..n)), n=0..7); # Peter Luschny, Oct 14 2011 MATHEMATICA T[n_, k_] := Binomial[n, k]^3*(k^2 + k + 1 + n^2 + n - k*n)/((k + 1)^2); Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018, after Peter Luschny *) PROG (C#)//code by Peter Luschny static int[] GenBinomial(int n, int k) {     int[, , ] T = new int[k, n, n ];     for (int m = 0; m < n; m++)     {         T[0, m, 0] = 1; T[0, m, m] = 1;         for (int j = 1; j < m; j++)         {             T[0, m, j] = T[0, m - 1, j] + T[0, m - 1, j - 1];         }         for (int r = 1; r < k; r++)         {             T[r, m, m] = 1;             for (int j = 0; j < m; j++)             {                 int p = (int)Math.Pow(T[0, m, j], r + 1);                 T[r, m, j] = p + T[0, m, j] * T[r - 1, m, m - j - 1];             }         }     }     int[] R = new int[n];     for (int j = 0; j < n; j++)     {         R[j] = T[k - 1, n - 1, j];     }     return R; } static int[] A194595_row(int r) {      return GenBinomial(r, 3); } // This C#-program causes numerical overflow for results // larger than 2147483647. - Susanne Wienand, Jun 25 2015 (PARI) A194595(n, k) = {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 CROSSREFS Cf. A103371, A197653, A197654, A197655, A197657, A007318, A002061, A000012, A027444, A073254. Sequence in context: A257852 A051927 A322069 * A219063 A218810 A218638 Adjacent sequences:  A194592 A194593 A194594 * A194596 A194597 A194598 KEYWORD nonn,tabl AUTHOR Susanne Wienand, Oct 10 2011 STATUS approved

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Last modified April 25 16:08 EDT 2019. Contains 322461 sequences. (Running on oeis4.)