A197655 Triangle by rows T(n,k), showing the number of meanders with length (n+1)*6 and containing (k+1)*6 Ls and (n-k)*6 Rs, where Ls and Rs denote arcs of equal length and a central angle of 60 degrees which are positively or negatively oriented.

1, 6, 1, 63, 126, 1, 364, 4374, 1092, 1, 1365, 85120, 127680, 5460, 1, 3906, 984375, 6000000, 1968750, 19530, 1, 9331, 7562646, 157828125, 210437500, 18906615, 55986, 1, 19608, 42824236, 2628749256, 11029593750, 4381248760, 128472708, 137256, 1

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 = 6.

The values in the triangle are proved by brute force for 0 <= n <= 5. The formulas are not yet proved in general.

The number triangle can be calculated recursively by the number triangles and A007318, A103371, A194595, A197653 and A197654. For n > 0, the first column seems to be A053700. The diagonal right hand is A000012. Row sums are in A198258.

The conjectured formulas are confirmed by dynamic programming for 0 <= n <= 19. - Susanne Wienand, Jul 04 2015

Links

Susanne Wienand, Table of n, a(n) for n = 0..209

Peter Luschny, Meanders and walks on the circle.

Formula

recursive formula (conjectured):

T(n,k) = T(6,n,k)

T(6,n,k) = T(1,n,k)^6 + T(1,n,k)*T(5,n,n-1-k), 0 <= k < n

T(6,n,n) = 1 k = n

T(5,n,k) = T(1,n,k)^5 + T(1,n,k)*T(4,n,n-1-k), 0 <= k < n

T(5,n,n) = 1 k = n

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(5,n,k) = A197654

T(4,n,k) = A197653

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 + E + F, k < n

A = (C(n,k))^6

B = (C(n,k))^5 * C(n,n-1-k)

C = (C(n,k))^4 *(C(n,n-1-k))^2

D = (C(n,k))^3 *(C(n,n-1-k))^3

E = (C(n,k))^2 *(C(n,n-1-k))^4

F = C(n,k) *(C(n,n-1-k))^5

[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,5). (Cf. A103371, A194595, A197653). [Peter Luschny, Oct 21 2011]

T(n,k) = A198065(n+1,k+1)C(n,k)^6/(k+1)^5. [Peter Luschny, Oct 29 2011]

T(n,k) = 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]

Example

For n = 4 and k = 2, T(n,k) = 127680
Example for recursive formula:
T(1,4,2) = 6
T(5,4,4-1-2) = T(5,4,1) = 13504
T(6,4,2) = 6^6 + 6*13504 = 127680
Example for closed formula:
T(4,2) = A + B + C + D + E + F
A = 6^6       =  46656
B = 6^5 * 4   =  31104
C = 6^4 * 4^2 =  20736
D = 6^3 * 4^3 =  13824
E = 6^2 * 4^4 =   9216
F = 6   * 4^5 =   6144
T(4,2)        = 127680
Some examples of list S and allocated values of dir if n = 4 and k = 2:
Length(S) = (4+1)*6 = 30 and S contains (2+1)*6 = 18 Ls.
  S: L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,R,R,R,R,R,R,R,R,R,R,R,R
dir: 1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,0,0,5,4,3,2,1,0,5,4,3,2,1
  S: L,L,L,L,L,L,L,L,L,L,R,L,L,R,L,R,R,R,L,R,R,L,R,R,R,L,L,R,R,L
dir: 1,2,3,4,5,0,1,2,3,4,4,4,5,5,5,5,4,3,3,3,2,2,2,1,0,0,1,1,0,0
  S: L,L,L,L,R,L,L,R,L,L,R,L,R,R,L,L,L,L,L,R,R,L,L,L,R,R,R,R,L,R
dir: 1,2,3,4,4,4,5,5,5,0,0,0,0,5,5,0,1,2,3,3,2,2,3,4,4,3,2,1,1,1
Each value of dir occurs 30/6 = 5 times.

Maple

A197655 := (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; seq(print(seq(A197655(n, k), k=0..n)), n=0..7); # Peter Luschny, Oct 21 2011

Mathematica

T[n_, k_] := (1 + n)(1 + 3k + 3k^2 - n - 3k*n + n^2)(1 + k + k^2 + n - k*n + n^2) Binomial[n, k]^6/(1 + k)^5;
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018, after Peter Luschny *)

Prog

(C#) static int[] A197655_row(int r) { return GenBinomial(r, 6); } // The function GenBinomial(r, s) is defined in A194595.
// This C#-program causes numerical overflow for results
// larger than 2147483647. - Susanne Wienand, Jul 04 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 A197655(n, k) : return S(5, n, k)
for n in (0..5) : print([A197655(n, k) for k in (0..n)])  # Peter Luschny, Oct 24 2011
(PARI)
A197655(n, k) = {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

Crossrefs

Cf. A000012, A007318, A053700, A103371, A194595, A197653, A197654, A198258.

Sequence in context: A352014 A292219 A364110 * A134279 A134280 A134278

Adjacent sequences: A197652 A197653 A197654 * A197656 A197657 A197658

Keyword

nonn,tabl

Author

Susanne Wienand, Oct 19 2011

Status

approved