login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A197654 Triangle by rows T(n,k), showing the number of meanders with length 5(n+1) and containing 5(k+1) L's and 5(n-k) R's, where L's and R's denote arcs of equal length and a central angle of 72 degrees which are positively or negatively oriented. 8
1, 5, 1, 31, 62, 1, 121, 1215, 363, 1, 341, 13504, 20256, 1364, 1, 781, 96875, 500000, 193750, 3905, 1, 1555, 501066, 7321875, 9762500, 1252665, 9330, 1, 2801, 2033647, 72656661, 262609375, 121094435, 6100941, 19607, 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 L's increment the index of dir,

(d) consecutive R's decrement the index of dir,

(e) the integer m>0 divides the length of S.

Then C is a meander if each value of dir occurs length(S)/m times.

Let T(m,n,k) = number of meanders (m, S, dir) in which S contains m(k+1) L's and m(n-k) R's, so that length(S) = m(n+1).

For this sequence, m = 5, T(n,k) = T(5,n,k).

The values in the triangle were proved by brute force for 0 <= n <= 6. The formulas have not yet been proved in general.

The number triangle can be calculated recursively by the number triangles and A007318, A103371, A194595 and A197653. The first column seems to be A053699.  The diagonal right hand is A000012. The diagonal with k = n-1 seems to be A152031 and to start with the second number of A152031. Row sums are in A198257.

The conjectured formulas are confirmed by dynamic programming for 0 <= n <= 29. - Susanne Wienand, Jul 01 2015

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

Peter Luschny, Meanders and walks on the circle.

Susanne Wienand, Example of a meander counted by A197654

FORMULA

Recursive formula (conjectured):

T(n,k) = 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(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,              k < n

                     A = (C(n,k))^5

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

                     C = (C(n,k))^3 *(C(n,n-1-k))^2

                     D = (C(n,k))^2 *(C(n,n-1-k))^3

                     E =  C(n,k)    *(C(n,n-1-k))^4     [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,4). [Peter Luschny, Oct 20 2011]

T(n,k) = A198064(n+1,k+1)C(n,k)^5/(k+1)^4. [Peter Luschny, Oct 29 2011]

T(n,k) = h(n,k)*binomial(n,k)^5, where h(n,k) = (1+k)*(1-((n-k)/(1+k))^5)/(1+2*k-n) if 1+2*k-n <> 0 else h(n,k) = 5. [Peter Luschny, Nov 24 2011]

EXAMPLE

For n = 5 and k = 2, T(n,k) = 500000

Example for recursive formula:

T(1,5,2) = 10

T(4,5,5-1-2) = T(4,5,2) = 40000

T(5,5,2) = 10^5 + 10*40000 = 500000

Example for closed formula:

T(5,2) = A + B + C + D + E

A = 10^5

B = 10^4 * 10

C = 10^3 * 10^2

D = 10^2 * 10^3

E = 10   * 10^4

T(5,2) = 5 * 10^5 = 500000

Some examples of list S and allocated values of dir if n = 5 and k = 2:

Length(S) = (5+1)*5 = 30 and S contains (2+1)*5 = 15 Ls.

  S: 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,R,R,R

dir: 1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,0,4,3,2,1,0,4,3,2,1,0,4,3,2,1

  S: L,L,L,L,L,L,L,R,R,L,L,R,R,R,L,R,R,R,L,R,L,L,L,R,R,L,R,R,R,R

dir: 1,2,3,4,0,1,2,2,1,1,2,2,1,0,0,0,4,3,3,3,3,4,0,0,4,4,4,3,2,1

  S: L,L,L,L,L,R,L,L,L,L,L,R,L,R,R,R,R,R,R,R,R,R,R,L,L,L,L,R,R,R

dir: 1,2,3,4,0,0,0,1,2,3,4,4,4,4,3,2,1,0,4,3,2,1,0,0,1,2,3,3,2,1

Each value of dir occurs 30/5 = 6 times.

The triangle begins:

1,

5, 1,

31, 62, 1,

121, 1215, 363, 1,

341, 13504, 20256, 1364, 1,

781, 96875, 500000, 193750, 3905, 1,

...

MAPLE

A197654 := (n, k)->(k^4+2*k^3*(1-n)+2*k^2*(2+n+2*n^2)+k*(3+n-n^2-3*n^3)+ n^4+n^3+n^2+n+1)*binomial(n, k)^5/(1+k)^4;

seq(print(seq(A197654(n, k), k=0..n)), n=0..7);  # Peter Luschny, Oct 20 2011

MATHEMATICA

T[n_, k_] := (k^4 + 2*k^3*(1-n) + 2*k^2*(2+n+2*n^2) + k*(3+n-n^2-3*n^3) + n^4+n^3+n^2+n+1)*Binomial[n, k]^5/(1+k)^4; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Feb 20 2017, after Peter Luschny *)

PROG

(C#) static int[] A197654_row(int r) { return GenBinomial(r, 5); } // The function GenBinomial(r, s) is defined in A194595.

// This C#-program causes numerical overflow for results larger than 2147483647. - Susanne Wienand, Jul 01 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 A197654(n, k) : return S(4, n, k)

for n in (0..5) : print [A197654(n, k) for k in (0..n)]  ## Peter Luschny, Oct 24 2011

(PARI) A197654(n, k) = {if(n ==1+2*k, 5, (1+k)*(1-((n-k)/(1+k))^5)/(1+2*k-n))*binomial(n, k)^5} \\ Peter Luschny, Nov 24 2011

CROSSREFS

Cf. A000012, A007318, A053699, A103371, A152031, A194595, A197653, A197655, A198257.

Sequence in context: A049353 A165226 A027759 * A296043 A066833 A039813

Adjacent sequences:  A197651 A197652 A197653 * A197655 A197656 A197657

KEYWORD

nonn,tabl

AUTHOR

Susanne Wienand, Oct 19 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 14 17:32 EST 2019. Contains 329979 sequences. (Running on oeis4.)