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A198257 Row sums of A197654. 6
1, 6, 94, 1700, 35466, 795312, 18848992, 464517468, 11801240050, 307073982116, 8147186436324, 219664321959524, 6003343077661216, 165975724832822400, 4634768975107569024, 130553813782898706908, 3705740233107582161538, 105902829964290241990332 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Number of meanders of length (n+1)*5 which are composed by arcs of equal length and a central angle of 72 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 = 5.
The terms are proved by brute force for 0 <= n <= 6, but not yet in general. [Susanne Wienand, Oct 29 2011]
LINKS
FORMULA
a(n) = Sum{k=0..n} Sum{j=0..4} Sum{i=0..4} (-1)^(j+i)*C(i,j)*C(n,k)^5*(n+1)^j*(k+1)^(4-j)/(k+1)^4. - Peter Luschny, Nov 02 2011
a(n) = Sum_{k=0..n} 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
a(n) ~ sqrt(5) * 2^(5*n+2) / (Pi*n)^2. - Vaclav Kotesovec, Apr 17 2023
EXAMPLE
Some examples of list S and allocated values of dir if n = 5:
Length(S) = (5+1)*5 = 30.
S: L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L
dir: 1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0
S: L,L,L,L,L,L,L,L,L,L,L,L,L,R,L,R,R,R,R,R,L,R,L,L,L,L,R,R,R,L
dir: 1,2,3,4,0,1,2,3,4,0,1,2,3,3,3,3,2,1,0,4,4,4,4,0,1,2,2,1,0,0
S: L,L,L,L,L,R,L,L,L,R,R,L,L,L,L,L,R,R,L,R,R,L,R,R,L,L,L,L,L,R
dir: 1,2,3,4,0,0,0,1,2,2,1,1,2,3,4,0,0,4,4,4,3,3,3,2,2,3,4,0,1,1
Each value of dir occurs 30/5 = 6 times.
MAPLE
A198257 := proc(n) local i, j, k, pow;
pow := (a, b) -> if a=0 and b=0 then 1 else a^b fi;
add(add(add((-1)^(j+i)*binomial(i, j)*binomial(n, k)^5*pow(n+1, j)*pow(k+1, 4-j)/(k+1)^4, i=0..4), j=0..4), k=0..n) end: seq(A198257(n), n=0..16); # Peter Luschny, Nov 02 2011
MATHEMATICA
Table[Sum[Sum[ Sum[(-1)^(j + i) Binomial[i, j], {i, 0, 4}] Binomial[n, k]^5*(n + 1)^j*(k + 1)^(4 - j), {j, 0, 4}]/(k + 1)^4, {k, 0, n}], {n, 0, 17}] (* Michael De Vlieger, Aug 18 2016 *)
PROG
(PARI)
A198257(n) = {sum(k=0, n, 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
Sequence in context: A078103 A221525 A321073 * A296820 A184983 A184980
KEYWORD
nonn
AUTHOR
Susanne Wienand, Oct 22 2011
STATUS
approved

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Last modified June 24 05:10 EDT 2024. Contains 373661 sequences. (Running on oeis4.)