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A198256 Row sums of A197653. 6
1, 5, 46, 485, 5626, 69062, 882540, 11614437, 156343330, 2142556130, 29791689148, 419260001030, 5960334608788, 85469709312860, 1234797737654296, 17955907741675749, 262607675818816050, 3860239468267647914, 57002176852356800700, 845159480056345448610 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of meanders of length (n+1)*4 which are composed by arcs of equal length and a central angle of 90 degrees.

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 terms are proved by brute force for 0 <= n <= 8, but not yet in general. [Susanne Wienand, Oct 29 2011]

LINKS

Table of n, a(n) for n=0..19.

Peter Luschny, Meanders and walks on the circle.

FORMULA

a(n) = Sum{k=0..n} Sum{j=0..3} Sum{i=0..3} (-1)^(j+i)*C(i,j)*C(n,k)^4*(n+1)^j*(k+1)^(3-j)/(k+1)^3. - Peter Luschny, Nov 02 2011

a(n) = Sum_{k=0..n} 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 else h(n,k) = 4. - Peter Luschny, Nov 24 2011

EXAMPLE

Some examples of list S and allocated values of dir if n = 4:

Length(S) = (4+1)*4 = 20.

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

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

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

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

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

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

Each value of dir occurs 20/4 = 5 times.

MATHEMATICA

A198256[n_] := Sum[Sum[Sum[(-1)^(j + i)* Binomial[i, j]*Binomial[n, k]^4*(n + 1)^j*(k + 1)^(3 - j)/(k + 1)^3, {i, 0, 3}], {j, 0, 3}], {k, 0, n}]; Table[A198256[n], {n, 0, 16}] (* Peter Luschny, Nov 02 2011 *)

PROG

(Sage) from mpmath import *

def A198256(n) : return hyper([1-n, 1-n, 1-n, 1-n], [3, 3, 3], 1)*(n^4-n^6)/4 + hyper([-n, -n, -n, -n], [2, 2, 2], 1)*(1+n+n^2+n^3) + hyper([2, 1-n, 1-n, 1-n, 1-n], [1, 3, 3, 3], 1)*(n^4+n^5)/4

mp.dps = 32

for n in (0..19) : print A198256(n)  ## Peter Luschny, Oct 24 2011

(PARI)

A198256(n) = {sum(k=0, n, 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. A198060, A198257, A198258.

Sequence in context: A042533 A309185 A169963 * A232972 A127304 A112029

Adjacent sequences:  A198253 A198254 A198255 * A198257 A198258 A198259

KEYWORD

nonn

AUTHOR

Susanne Wienand, Oct 22 2011

STATUS

approved

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Last modified December 8 12:07 EST 2019. Contains 329862 sequences. (Running on oeis4.)