A076876 -id:A076876 - cp's OEIS frontend

A210592 Meandric numbers for a river crossing up to 19 parallel roads at n points.

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123439, 5964691, 19302316, 54898417, 179696558, 516468943, 1707136794, 4950706511, 16503342067, 48232628420, 161984027241, 476636439213, 1611286734027, 4769639146736, 16218693724179

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 19 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=19 of A380367.
Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432 (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21) onwards from Andrew Howroyd, Jan 31 2025

A380367 Array read by antidiagonals: meandric numbers for a river crossing up to k parallel roads at n points, n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 4, 8, 8, 1, 1, 2, 4, 9, 14, 14, 1, 1, 2, 4, 10, 21, 43, 42, 1, 1, 2, 4, 10, 22, 52, 81, 81, 1, 1, 2, 4, 10, 23, 61, 131, 272, 262, 1, 1, 2, 4, 10, 23, 62, 142, 345, 538, 538, 1, 1, 2, 4, 10, 23, 63, 153, 420, 915, 1920, 1828

Offset: 0

Andrew Howroyd, Jan 31 2025

Illustrations of the initial terms for the case of two parallel roads can be found in A076876.

			Array begins:
===================================================
n\k |   1   2   3    4    5    6    7    8    9 ...
----+----------------------------------------------
  0 |   1   1   1    1    1    1    1    1    1 ...
  1 |   1   1   1    1    1    1    1    1    1 ...
  2 |   1   2   2    2    2    2    2    2    2 ...
  3 |   2   3   4    4    4    4    4    4    4 ...
  4 |   3   8   9   10   10   10   10   10   10 ...
  5 |   8  14  21   22   23   23   23   23   23 ...
  6 |  14  43  52   61   62   63   63   63   63 ...
  7 |  42  81 131  142  153  154  155  155  155 ...
  8 |  81 272 345  420  433  446  447  448  448 ...
  9 | 262 538 915 1017 1120 1135 1150 1151 1152 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..945 (first 43 antidiagonals)

Columns 1..19 are A005316, A076876, A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592.
Main diagonal is A206432.
Cf. A076875 (perpendicular roads).

T(n,k) = T(n,n) for k > n.

A107321 a(n)=Sum(i+j+k+l+...+r+s=n) A005316(i)A005316(j)...*A005316(s) and the ordered partition of n runs over all odd i, all even j,k,.., r, all i,j,...,r,s>=1.

Original entry on oeis.org

1, 1, 2, 3, 8, 14, 42, 79, 254, 506, 1702, 3548, 12320, 26666, 94794, 211751, 766362, 1758352, 6453812, 15150922, 56238710, 134659120

Offset: 0

R. J. Mathar, May 06 2006

A lower bound to A076876. a(n) counts the cases where all crossings k are East of the crossings i, all crossings l East of the crossings j etc. Some backwinding interlaced crossings are counted in A076876(n) but not here.

A124495 G.f.: A(x) = 1/[1-x - Sum_{n>=1} A001147(n)x^(2n) ] where A001147(n) = (2n)!/(n!2^n) is the double factorials.

Original entry on oeis.org

1, 1, 2, 3, 8, 14, 43, 81, 283, 556, 2243, 4512, 21374, 43469, 243817, 497217, 3289606, 6697795, 51583952, 104698998, 922789643, 1867079621, 18522929815, 37380015420, 411572179999, 828925168492, 10014624164666, 20140445929353

Offset: 0

Paul D. Hanna, Nov 04 2006

nonn

Is this sequence equal to A076876 (meandric numbers for a river crossing two parallel roads at n points)?

			G.f.: A(x) = 1/(1-x - x^2 - 3*x^4 - 15*x^6 - 105*x^8 - 945*x^10 -...).

Cf. A001147.

a(n)=polcoeff(1/(1-x-sum(k=1,n\2,(2*k)!/k!/2^k*x^(2*k))+x*O(x^n)),n)

cp's OEIS Frontend

A210592 Meandric numbers for a river crossing up to 19 parallel roads at n points.

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A380367 Array read by antidiagonals: meandric numbers for a river crossing up to k parallel roads at n points, n >= 0, k >= 1.

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Examples

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Formula

A107321 a(n)=Sum(i+j+k+l+...+r+s=n) A005316(i)A005316(j)...*A005316(s) and the ordered partition of n runs over all odd i, all even j,k,.., r, all i,j,...,r,s>=1.

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Author

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Comments

A124495 G.f.: A(x) = 1/[1-x - Sum_{n>=1} A001147(n)x^(2n) ] where A001147(n) = (2n)!/(n!2^n) is the double factorials.

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PARI

cp's OEIS Frontend

A210592 Meandric numbers for a river crossing up to 19 parallel roads at n points.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A380367 Array read by antidiagonals: meandric numbers for a river crossing up to k parallel roads at n points, n >= 0, k >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A107321 a(n)=Sum(i+j+k+l+...+r+s=n) A005316(i)*A005316(j)*...*A005316(s) and the ordered partition of n runs over all odd i, all even j,k,.., r, all i,j,...,r,s>=1.

Views

Author

Keywords

Comments

A124495 G.f.: A(x) = 1/[1-x - Sum_{n>=1} A001147(n)*x^(2n) ] where A001147(n) = (2n)!/(n!*2^n) is the double factorials.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A107321 a(n)=Sum(i+j+k+l+...+r+s=n) A005316(i)A005316(j)...*A005316(s) and the ordered partition of n runs over all odd i, all even j,k,.., r, all i,j,...,r,s>=1.

A124495 G.f.: A(x) = 1/[1-x - Sum_{n>=1} A001147(n)x^(2n) ] where A001147(n) = (2n)!/(n!2^n) is the double factorials.