A103327 -id:A103327 - cp's OEIS frontend

A111195 a(n) = 2^(-n) * Sum_{k=0..n} binomial(2n+1, 2k+1) * A000364(k).

1, 2, 5, 26, 269, 4666, 121017, 4370722, 209364537, 12833657010, 979336390669, 91018760056938, 10120101446389765, 1326280083965014634, 202311875122389093761, 35535622109342844729074

Offset: 0

Philippe Deléham, Oct 24 2005

Cf. A000364, A103327, A308715.

t = Range[0, 34]!CoefficientList[ Series[ Sec[x], {x, 0, 34}], x]; f[n_] := 2^(-n)*Sum [Binomial[2n + 1, 2k + 1]*t[[2k + 1]], {k, 0, n}]; Table[ f[n], {n, 0, 17}] (* Robert G. Wilson v, Oct 24 2005 *)
Table[Sum[Binomial[2*n + 1, 2*k + 1]*Abs[EulerE[2*k]], {k, 0, n}] / 2^n, {n, 0, 20}] (* Vaclav Kotesovec, Jul 10 2021 *)

a(n) ~ cosh(Pi/2) * 2^(3*n + 3) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Jul 10 2021

More terms from Robert G. Wilson v, Oct 24 2005

A111196 a(n) = 2^(-n)Sum_{k=0..n} binomial(2n+1, 2k+1)A000364(n-k).

Original entry on oeis.org

1, 2, 9, 78, 1141, 25442, 804309, 34227438, 1886573641, 130746521282, 11127809595009, 1141012634368398, 138730500808639741, 19735099323279743522, 3247323803322747092109, 611982206046097666022958

Offset: 0

Philippe Deléham, Oct 24 2005

Cf. A103327, A002084, A000364.

t = Range[0, 32]!CoefficientList[ Series[ Sec[x], {x, 0, 32}], x]; f[n_] := 2^(-n)*Sum [Binomial[2n + 1, 2k + 1]*t[[2n - 2k + 1]], {k, 0, n}]; Table[ f[n], {n, 0, 16}] (* Robert G. Wilson v, Oct 24 2005 *)
Table[Sum[Binomial[2*n + 1, 2*k + 1]*Abs[EulerE[2*(n-k)]], {k, 0, n}] / 2^n, {n, 0, 20}] (* Vaclav Kotesovec, Jul 10 2021 *)

a(n) = 2^(-n)*A002084(n).

a(n) ~ sinh(Pi/2) * 2^(3*n + 5) * n^(2*n + 3/2) / (Pi^(2*n + 3/2) * exp(2*n)). - Vaclav Kotesovec, Jul 10 2021

More terms from Robert G. Wilson v, Oct 24 2005

A361894 Triangle read by rows. T(n, k) is the number of Fibonacci meanders with a central angle of 360/m degrees that make mk left turns and whose length is mn, where m = 2.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 6, 2, 1, 5, 16, 6, 2, 1, 6, 35, 20, 6, 2, 1, 7, 66, 65, 20, 6, 2, 1, 8, 112, 186, 70, 20, 6, 2, 1, 9, 176, 462, 246, 70, 20, 6, 2, 1, 10, 261, 1016, 812, 252, 70, 20, 6, 2, 1, 11, 370, 2025, 2416, 917, 252, 70, 20, 6, 2, 1, 12, 506, 3730, 6435, 3256, 924, 252, 70, 20, 6, 2, 1

Offset: 1

Peter Luschny, Mar 31 2023

For an overview of the terms used see A361574. A201631 gives the row sums of this triangle.
The corresponding sequence counting meanders without the requirement of being Fibonacci is A103371 (for which in turn A103327 is a termwise majorant counting permutations of the same type).
The diagonals, starting from the main diagonal, apparently converge to A000984.

			Triangle T(n, k) starts:
  [ 1]  1;
  [ 2]  2,   1;
  [ 3]  3,   2,    1;
  [ 4]  4,   6,    2,    1;
  [ 5]  5,  16,    6,    2,    1;
  [ 6]  6,  35,   20,    6,    2,   1;
  [ 7]  7,  66,   65,   20,    6,   2,   1;
  [ 8]  8, 112,  186,   70,   20,   6,   2,  1;
  [ 9]  9, 176,  462,  246,   70,  20,   6,  2,  1;
  [10] 10, 261, 1016,  812,  252,  70,  20,  6,  2, 1;
  [11] 11, 370, 2025, 2416,  917, 252,  70, 20,  6, 2, 1;
  [12] 12, 506, 3730, 6435, 3256, 924, 252, 70, 20, 6, 2, 1.
.
T(4, k) counts Fibonacci meanders with central angle 180 degrees and length 8 that make k left turns. Written as binary strings (L = 1, R = 0):
k = 1: 11000000, 10010000, 10000100, 10000001;
k = 2: 11110000, 11100100, 11100001, 11010010, 11001001, 10100101;
k = 3: 11111100, 11111001;
k = 4: 11111111.

Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Enumeration of Dyck paths with air pockets, arXiv:2202.06893 [cs.DM], 2022-2023.
Peter Luschny, Fibonacci meanders.

Cf. A201631 (row sums), A361681 (m=3), A132812, A361574, A103371, A000984.

# using function 'FibonacciMeandersByLeftTurns' from A361681.
for n in range(1, 12):
    print(FibonacciMeandersByLeftTurns(2, n))

cp's OEIS Frontend

A111195 a(n) = 2^(-n) * Sum_{k=0..n} binomial(2n+1, 2k+1) * A000364(k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions

A111196 a(n) = 2^(-n)Sum_{k=0..n} binomial(2n+1, 2k+1)A000364(n-k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions

A361894 Triangle read by rows. T(n, k) is the number of Fibonacci meanders with a central angle of 360/m degrees that make mk left turns and whose length is mn, where m = 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

SageMath

cp's OEIS Frontend

A111195 a(n) = 2^(-n) * Sum_{k=0..n} binomial(2*n+1, 2*k+1) * A000364(k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions

A111196 a(n) = 2^(-n)*Sum_{k=0..n} binomial(2*n+1, 2*k+1)*A000364(n-k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions

A361894 Triangle read by rows. T(n, k) is the number of Fibonacci meanders with a central angle of 360/m degrees that make m*k left turns and whose length is m*n, where m = 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

SageMath

A111195 a(n) = 2^(-n) * Sum_{k=0..n} binomial(2n+1, 2k+1) * A000364(k).

A111196 a(n) = 2^(-n)Sum_{k=0..n} binomial(2n+1, 2k+1)A000364(n-k).

A361894 Triangle read by rows. T(n, k) is the number of Fibonacci meanders with a central angle of 360/m degrees that make mk left turns and whose length is mn, where m = 2.