A198060 -id:A198060 - cp's OEIS frontend

A198256 Row sums of A197653.

1, 5, 46, 485, 5626, 69062, 882540, 11614437, 156343330, 2142556130, 29791689148, 419260001030, 5960334608788, 85469709312860, 1234797737654296, 17955907741675749, 262607675818816050, 3860239468267647914, 57002176852356800700, 845159480056345448610

Offset: 0

Susanne Wienand, Oct 22 2011

nonn

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

			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.

Peter Luschny, Meanders and walks on the circle.

Cf. A197653, A198060, A198257, A198258.
Cf. A005260, A181069.
Cf. A181067, A374614.

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 *)

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

from mpmath import mp, hyper
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(int(A198256(n)))  # Peter Luschny, Oct 24 2011

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
From Peter Bala, Mar 21 2023: (Start)
Conjecture 1: a(n) = Sum_{k = 0..n} binomial(n+1,k)^2*binomial(n,k)^2.
If true, then we have the third-order recurrence equation
n^2*(n + 1)^3*P(n-1)*a(n) = 2*n^2*(400*n^8 - 1260*n^7 + 20*n^6 + 3020*n^5 - 1646*n^4 - 1951*n^3 + 1142*n^2 + 465*n - 290)*a(n-1) + 4*(n - 1)*(2800*n^9 - 15420*n^8 + 30620*n^7 - 23710*n^6 + 808*n^5 + 6863*n^4 - 1309*n^3 - 1218*n^2 + 496*n - 60)*a(n-2) + 8*(n - 2)^2*(2*n - 3)*(4*n - 5)*(4*n - 7)*P(n)*a(n-3) with a(0) = 1, a(1) = 5 and a(2) = 46 and where P(n) = 100*n^5 - 65*n^4 - 35*n^3 + 25*n^2 + 6*n - 5.
Conjecture 2: working with offset 1, that is, a(1) = 1, a(2) = 5, ..., then the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for positive integers n and r and all primes p >= 5. (End)
a(n) ~ 2^(4*n + 5/2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Apr 17 2023
Peter Bala's conjecture 1 can equivalently written: a(n) = hypergeom([-n - 1, -n - 1, -n, -n], [1, 1, 1], 1). - Detlef Meya, May 28 2024
a(n) = Sum_{k=0..n+1} (k/(n+1))^2 * binomial(n+1,k)^4. - Seiichi Manyama, Jul 14 2024

A197657 Row sums of A194595.

Original entry on oeis.org

1, 4, 22, 134, 866, 5812, 40048, 281374, 2006698, 14482064, 105527060, 775113440, 5731756720, 42628923040, 318621793472, 2391808860446, 18023208400634, 136271601087352, 1033449449559724, 7858699302115444, 59906766929537116, 457685157123172664

Offset: 0

Susanne Wienand, Oct 17 2011

Number of meanders of length (n+1)*3 which are composed by arcs of equal length and a central angle of 120 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 = 3.
For 0 <= n <= 16, a(n) = the hypergraph Fuss-Catalan number FC_1^(2,n+1) in the notation of Chavan et al. - see 7.1 in the Appendix. - Peter Bala, Apr 11 2023

			Some examples of list S and allocated values of dir if n = 4:
Length(S) = (4+1)*3 = 15.
  S: L,L,L,L,L,L,L,L,L,L,L,L,L,L,L
dir: 1,2,0,1,2,0,1,2,0,1,2,0,1,2,0
  S: L,L,L,L,R,L,L,R,L,L,R,L,L,L,L
dir: 1,2,0,1,1,1,2,2,2,0,0,0,1,2,0
  S: L,R,L,L,L,L,L,R,L,L,R,L,R,R,R
dir: 1,1,1,2,0,1,2,2,2,0,0,0,0,2,1
Each value of dir occurs 15/3 = 5 times.

Parth Chavan, Andrew Lee and Karthik Seetharaman, Hypergraph Fuss-Catalan numbers, arXiv:2202.01111 [math.CO]
Peter Luschny, Meanders and walks on the circle.
Susanne Wienand, Animation of a meander
Susanne Wienand, Example of a meander

Cf. A198060, A198256, A198257, A198258.

A197657 := proc(n)
    (A000172(n) + A000172(n+1)) / 3 ;
end proc; # R. J. Mathar, Jul 26 2014
a := n -> 2^n*hypergeom([n + 1, -n/2, -n/2 - 1/2], [1, 1], 1):
seq(simplify(a(n)), n = 0..21); # Peter Luschny, Mar 26 2023

A197657[n_] := Sum[Sum[Sum[(-1)^(j + i)* Binomial[i, j]*Binomial[n, k]^3*(n + 1)^j*(k + 1)^(2 - j)/(k + 1)^2, {i, 0, 2}], {j, 0, 2}], {k, 0, n}]; Table[A197657[n], {n, 0, 16}]  (* Peter Luschny, Nov 02 2011 *)

A197657(n) = {sum(k=0,n,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

def A197657(n):
    return 2^n*hypergeometric([n + 1, -n/2, -n/2 - 1/2], [1, 1], 1).simplify_hypergeometric()
for n in (0..21): print(A197657(n)) # Peter Luschny, Mar 26 2023

a(n) = Sum{k=0..n} Sum{j=0..2} Sum{i=0..2} (-1)^(j+i)*C(i,j)*C(n,k)^3*(n+1)^j*(k+1)^(2-j)/(k+1)^2. - Peter Luschny, Nov 02 2011
a(n) = Sum_{k=0..n} 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
a(n) = A141147(n+1)/2 = A110707(n+1)/6 = (A000172(n)+A000172(n+1))/3. - Max Alekseyev, Jul 15 2014
Conjecture: (n+1)^2*a(n) -3*(n+1)*(2*n+1)*a(n-1) -3*n*(5*n-7)*a(n-2) -8*(n-2)^2*a(n-3)=0. - R. J. Mathar, Jul 26 2014
a(n) = 2^n*hypergeom([n + 1, -n/2, -n/2 - 1/2], [1, 1], 1). - Peter Luschny, Mar 26 2023
a(n) ~ sqrt(3) * 2^(3*n+1) / (Pi*n). - Vaclav Kotesovec, Apr 17 2023

A198257 Row sums of A197654.

Original entry on oeis.org

1, 6, 94, 1700, 35466, 795312, 18848992, 464517468, 11801240050, 307073982116, 8147186436324, 219664321959524, 6003343077661216, 165975724832822400, 4634768975107569024, 130553813782898706908, 3705740233107582161538, 105902829964290241990332

Offset: 0

Susanne Wienand, Oct 22 2011

nonn

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]

			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.

Peter Luschny, Meanders and walks on the circle.
Project Euler, Robot Walks: Problem 208

Cf. A198060, A198256, A198258.

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

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 *)

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

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

A198258 Row sums of A197655.

Original entry on oeis.org

1, 7, 190, 5831, 219626, 8976562, 394800204, 18211045575, 873216720082, 43136486269382, 2183722698469676, 112795257850321202, 5925951358743662260, 315869014732813229716, 17048301919464100932440, 930217336628892162216135, 51244644190683748419791970

Offset: 0

Susanne Wienand, Oct 24 2011

nonn

Number of meanders of length (n+1)*6 which are composed by arcs of equal length and a central angle of 60 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 = 6.
The terms are proved by brute force for 0 <= n <= 5, but not yet in general. - Susanne Wienand, Oct 29 2011

			Some examples of list S and allocated values of dir if n = 4:
Length(S) = (4+1)*6 = 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: 1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,0
  S: L,L,L,L,L,L,L,R,R,R,R,L,R,R,R,R,R,L,L,L,L,R,R,L,L,L,L,R,L,L
dir: 1,2,3,4,5,0,1,1,0,5,4,4,4,3,2,1,0,0,1,2,3,3,2,2,3,4,5,5,5,0
  S: L,L,L,L,L,R,R,R,R,R,R,R,L,L,R,L,L,R,L,L,R,L,L,L,R,R,L,L,L,L
dir: 1,2,3,4,5,5,4,3,2,1,0,5,5,0,0,0,1,1,1,2,2,2,3,4,4,3,3,4,5,0
Each value of dir occurs 30/6 = 5 times.

Peter Luschny, Meanders and walks on the circle.

Cf. A198060, A198256, A198257.

A198258 := 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)^6*pow(n+1,j)*pow(k+1,5-j)/(k+1)^5, i=0..5),j=0..5),k=0..n) end:
seq(A198258(n), n=0..16); # Peter Luschny, Nov 02 2011

T[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;
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jun 29 2019 *)

A198258(n) = {sum(k=0,n,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

a(n) = Sum_{k=0..n} Sum_{j=0..5} Sum_{i=0..5} (-1)^(j+i)*C(i,j)*C(n,k)^6*(n+1)^j*(k+1)^(5-j)/(k+1)^5. - Peter Luschny, Nov 02 2011
a(n) = Sum_{k=0..n} 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
Conjecture: working with offset 1, that is, a(1) = 1, a(2) = 7, ..., then the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for positive integers n and r and all primes p >= 3. - Peter Bala, Mar 21 2023
a(n) ~ sqrt(3) * 2^(6*n+3) / (Pi*n)^(5/2). - Vaclav Kotesovec, Apr 17 2023

A199932 Number of meanders of length n.

Original entry on oeis.org

1, 3, 5, 12, 17, 47, 65, 169, 279, 645, 1025, 2698, 4097, 9917, 17345, 39698, 65537, 161395, 262145, 624004, 1089007, 2449881, 4194305, 10097733, 16812683, 38754747, 69117097, 155178266, 268435457, 629929761, 1073741825, 2459703907, 4400500499, 9756737721

Offset: 1

Peter Luschny, Nov 20 2011

nonn

A meander is a closed curve drawn by arcs of equal length and central angles of equal magnitude, starting with a positively oriented arc.

a(n) = 2^(n-1) + 1 iff n is prime.

Peter Luschny, Meander.

Cf. A198060, A200062.

A199932 := proc(n) local d, k, j, i; add(add(add(add(
(-1)^(j+i)*binomial(i,j)*binomial(n/d-1,k)^d*((n/d)/(k+1))^j,
i=0..d-1),j=0..d-1),k=0..(n/d-1)),d=numtheory[divisors](n)) end:
seq(A199932(i),i=1..34);

A198060[m_, n_] := Sum[ Sum[ Sum[(-1)^(j+i)*Binomial[i, j]* Binomial[n, k]^(m+1)*(n+1)^j*(k+1)^(m-j)/(k+1)^m, {i, 0, m}], {j, 0, m}], {k, 0, n}]; a[n_] := Sum[ A198060[d-1, n/d-1], {d, Divisors[n]}]; Table[a[n], {n, 1, 34}] (* Jean-François Alcover, Jun 27 2013 *)

a(n) = Sum_{d|n} A198060(d-1,n/d-1).

A198061 Array read by antidiagonals, m>=0, n>=0, A(m,n) = sum{k=0..n} sum{j=0..m} sum{i=0..m} (-1)^(j+i)C(i,j)n^j*k^(m-j).

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 0, 2, 6, 4, 0, 2, 11, 12, 5, 0, 2, 20, 32, 20, 6, 0, 2, 37, 84, 70, 30, 7, 0, 2, 70, 224, 240, 130, 42, 8, 0, 2, 135, 612, 834, 550, 217, 56, 9, 0, 2, 264, 1712, 2968, 2354, 1092, 336, 72, 10, 0, 2, 521, 4884, 10826, 10310, 5551, 1960, 492

Offset: 0

Peter Luschny, Nov 02 2011

			m\n  [0] [1]  [2]   [3]    [4]     [5]    [6]
----------------------------------------------
[0]   1   2    3     4      5       6       7    A000027
[1]   0   2    6    12     20      30      42    A002378
[2]   0   2   11    32     70     130     217    A033994
[3]   0   2   20    84    240     550    1092    A098077
[4]   0   2   37   224    834    2354    5551
[5]   0   2   70   612   2968   10310   28854

Cf. A198060.

A198061 := proc(m, 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)*pow(n,j)*pow(k,m-j),i=0..m),j=0..m),k=0..n) end:
for m from 0 to 8 do lprint(seq(A198061(m,n), n=0..6)) od;

Unprotect[Power]; 0^0 = 1; Protect[Power]; a[m_, n_] :=  Sum[(-1)^(j+i)*Binomial[i, j]*n^j*k^(m-j) , {i, 0, m}, {j, 0, m}, {k, 0, n}]; Table[a[m-n, n], {m, 0, 10}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jul 26 2013 *)

A198061(n,2) = A006127(n+1)

A200062 Meanders of length n and central angle < 360 degrees.

Original entry on oeis.org

0, 1, 1, 4, 1, 15, 1, 41, 23, 133, 1, 650, 1, 1725, 961, 6930, 1, 30323, 1, 99716, 40431, 352729, 1, 1709125, 35467, 5200315, 2008233, 20960538, 1, 93058849, 1, 312220259, 105533203, 1166803129, 20194059, 5478229800, 1, 17672631921, 5731781295, 71539226243, 1

Offset: 1

Peter Luschny, Nov 16 2011

nonn

A meander is a closed curve drawn by arcs of equal length and central angles of equal magnitude, starting with a positively oriented arc.

a(n) = 1 if and only if n is prime.

			See the link for n = 6,8,9.

Peter Luschny, Table of n, a(n) for n = 1..1000
Peter Luschny, Meander.

Cf. A198060, A199932.

A200062 := proc(n) local i;
add(A198060(i-1,n/i-1),i=numtheory[divisors](n)) - 2^(n-1) end: seq(A200062(i),i=1..41);

A198060[m_, n_] := Sum[ Sum[ Sum[ (-1)^(j+i)*Binomial[i, j]*Binomial[n, k]^(m+1) * (n+1)^j * (k+1)^(m-j) / (k+1)^m, {i, 0, m}], {j, 0, m}], {k, 0, n}]; a[n_] := Sum[ A198060[d-1, n/d-1], {d, Divisors[n]}] - 2^(n-1); Table[a[n], {n, 1, 41}] (* Jean-François Alcover, Jun 27 2013 *)

A200062(n) = { D = divisors(n);
sum(m = 2, #D, d = D[m];
   sum(k=0,n/d-1,binomial(n/d-1,k)^d*
      sum(j=0,d-1,((n/d)/(k+1))^j*
          sum(i=0,d-1,(-1)^(j+i)*binomial(i,j)
))))}

a(n) = Sum_{d|n} A198060(d-1,n/d-1) - 2^(n-1).

A198062 Array read by antidiagonals, m>=0, n>=0, k>=0, A(m, n, k) = sum{j=0..m} sum{i=0..m} (-1)^(j+i)C(i,j)n^j*k^(m-j).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 4, 2, 1, 0, 1, 1, 8, 3, 2, 1, 0, 1, 1, 16, 4, 4, 3, 1, 0, 1, 1, 32, 5, 8, 9, 3, 1, 0, 1, 1, 64, 6, 16, 27, 7, 3, 1, 0, 1, 1, 128, 7, 32, 81, 15, 7, 3, 1, 0, 1, 1, 256, 8, 64, 243, 31, 15, 9, 4, 1, 0, 1, 1, 512, 9

Offset: 0

Peter Luschny, Nov 02 2011

			   [0] [1] [2]  [3] [4]  [5]  [6]  [7]  [8]  [9]
-------------------------------------------------
[0]  1   1   1    1   1    1    1    1    1    1    A000012
[1]  0   1   1    2   2    2    3    3    3    3    A003056
[2]  0   1   1    4   3    4    9    7    7    9    A073254
[3]  0   1   1    8   4    8   27   15   15   27    A198063
[4]  0   1   1   16   5   16   81   31   31   81    A198064
[5]  0   1   1   32   6   32  243   63   63  243    A198065

Vincenzo Librandi, Table of n, a(n) for n = 0..350

Cf. A198060, A198061.

A198062_RowAsTriangle := proc(m) local pow; pow :=(a,b)->`if`(a=0 and b=0,1,a^b): proc(n, k) local i, j; add(add((-1)^(j + i)*binomial(i, j)*pow(n, j)* pow(k, m-j), i=0..m),j=0..m) end: end:
for m from 0 to 2 do seq(print(seq(A198062_RowAsTriangle(m)(n,k),k=0..n)),n=0..5) od;

max = 9; RowAsTriangle[m_][n_, k_] := Module[{pow}, pow[a_, b_] := If[a == 0 && b == 0, 1, a^b]; Module[{i, j}, Sum[Sum[(-1)^(j+i)*Binomial[i, j]*pow[n, j]*pow[k, m-j], {i, 0, m}], {j, 0, m}]]]; t = Flatten /@ Table[RowAsTriangle[m][n, k], {m, 0, max}, {n, 0, max}, {k, 0, n}]; Table[t[[n-k+1, k+1]], {n, 0, max}, {k, 0, n }] // Flatten (* Jean-François Alcover, Feb 25 2014, after Maple *)

A007318(n,k) = A(0,n+1,k+1)*C(n,k)^1/(k+1)^0,
A103371(n,k) = A(1,n+1,k+1)*C(n,k)^2/(k+1)^1,
A194595(n,k) = A(2,n+1,k+1)*C(n,k)^3/(k+1)^2,
A197653(n,k) = A(3,n+1,k+1)*C(n,k)^4/(k+1)^3,
A197654(n,k) = A(4,n+1,k+1)*C(n,k)^5/(k+1)^4,
A197655(n,k) = A(5,n+1,k+1)*C(n,k)^6/(k+1)^5.

A200583 Table read by rows, n >= 1, 1 <= k <= card(divisors(n)), T(n,k) meanders of length n and central angle of 360/d degrees, d the k-th divisor of n.

Original entry on oeis.org

1, 2, 1, 4, 1, 8, 3, 1, 16, 1, 32, 10, 4, 1, 64, 1, 128, 35, 5, 1, 256, 22, 1, 512, 126, 6, 1, 1024, 1, 2048, 462, 134, 46, 7, 1, 4096, 1, 8192, 1716, 8, 1, 16384, 866, 94, 1, 32768, 6435, 485, 9, 1, 65536, 1, 131072, 24310, 5812, 190, 10, 1, 262144, 1, 524288

Offset: 1

Peter Luschny, Nov 20 2011

A meander is a closed curve drawn by arcs of equal length and central angle of equal magnitude, starting with a positively oriented arc.

			[ 1]            1
[ 2]          2, 1
[ 3]          4, 1
[ 4]        8, 3, 1
[ 5]         16, 1
[ 6]      32, 10, 4, 1
[ 7]         64, 1
[ 8]     128, 35, 5, 1
[ 9]       256, 22, 1
[10]     512, 126, 6, 1
[11]        1024, 1
[12] 2048, 462, 134, 46, 7, 1

Vincenzo Librandi, Table of n, a(n) for n = 1..482
Peter Luschny, Meander.

Cf. A198060, A199932, A200062.

A200583_row := proc(n) local i;
seq(A198060(i-1,n/i-1),i=numtheory[divisors](n)) end:
seq(print(A200583_row(i)),i=1..12);

A198060[m_, n_] := Sum[Sum[Sum[(-1)^(j+i)*Binomial[i, j]*Binomial[n, k]^(m+1)*(n+1)^j*(k+1)^(m-j)/(k+1)^m, {i, 0, m}], {j, 0, m}], {k, 0, n}]; row[n_] := Table[A198060[d-1, n/d-1], {d, Divisors[n]}]; Table[row[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Feb 25 2014, after Maple *)

T(n,k) = A198060(d-1,n/d-1) where d is the k-th divisor of n (the divisors in natural order).

A361575 Number of Fibonacci meanders of length n.

Original entry on oeis.org

1, 3, 5, 11, 13, 30, 34, 71, 97, 177, 233, 481, 610, 1157, 1677, 3027, 4181, 8016, 10946, 20379, 29534, 52461, 75025, 140748, 196778, 355979, 526123, 933044, 1346269, 2469992, 3524578, 6342729, 9400985, 16487211

Offset: 1

Peter Luschny, Mar 16 2023

For an overview of the terms and functions used, compare A361574. The corresponding sequence counting meanders without the requirement to be Fibonacci is A199932.

			Fibonacci meanders with length 6 can have the central angle 360/m, where m is in divisors(6) = {1, 2, 3, 6}. In total there are a(6) = 30 such meanders, the list shows their binary representation together with the multiplicity with which they appear.
100000 x 1, 100001 x 2, 100010 x 1, 100100 x 2, 100101 x 1, 101000 x 1,
101001 x 1, 101010 x 1, 110000 x 2, 110001 x 2, 110010 x 1, 110100 x 1,
110101 x 1, 111000 x 2, 111001 x 2, 111010 x 1, 111100 x 2, 111101 x 1,
111110 x 1, 111111 x 4.

Peter Luschny, Fibonacci meanders.

Cf. A361574, A199932, A198060.

# The list A was computed with the functions given in A361574. They correspond to the columns in the table shown in the reference.
A := [[1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596, 2583, 4180, 6764, 10945, 17710, 28656, 46367, 75024, 121392, 196417, 317810, 514228, 832039, 1346268, 2178308, 3524577, 5702886, 9227464, 14930351], [1, 3, 6, 13, 30, 70, 167, 405, 992, 2450, 6090, 15214, 38165, 96069, 242530, 613811, 1556856], [1, 3, 8, 21, 68, 242, 861, 3151, 11874, 45192, 173496], [1, 3, 10, 35, 154, 858, 4723, 25625], [1, 3, 12, 61, 360, 3058], [1, 3, 14, 111, 878], [1, 3, 16, 209], [1, 3, 18, 403], [1, 3, 20], [1, 3, 22], [1, 3, 24], [1, 3], [1, 3], [1, 3], [1, 3], [1, 3], [1, 3], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1]];
with(LinearAlgebra):  # a(n) is the sum of row n of this table.
row := k -> [seq(`if`(irem(n, k) <> 0, 0, A[k][n/k]), n = 1..34)]:
M := Transpose(Matrix([seq(row(n), n = 1..34)])):
seq(add(m, m = Row(M, n)), n = 1..34);

cp's OEIS Frontend

A198256 Row sums of A197653.

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A197657 Row sums of A194595.

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A198257 Row sums of A197654.

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A198258 Row sums of A197655.

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A199932 Number of meanders of length n.

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A198061 Array read by antidiagonals, m>=0, n>=0, A(m,n) = sum{k=0..n} sum{j=0..m} sum{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j).

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A200062 Meanders of length n and central angle < 360 degrees.

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A198061 Array read by antidiagonals, m>=0, n>=0, A(m,n) = sum{k=0..n} sum{j=0..m} sum{i=0..m} (-1)^(j+i)C(i,j)n^j*k^(m-j).

A198062 Array read by antidiagonals, m>=0, n>=0, k>=0, A(m, n, k) = sum{j=0..m} sum{i=0..m} (-1)^(j+i)C(i,j)n^j*k^(m-j).