A000891 -id:A000891 - cp's OEIS frontend

A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)binomial(2k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.

0, 0, 1, 0, 1, 3, 0, 1, 8, 20, 0, 1, 15, 75, 175, 0, 1, 24, 189, 784, 1764, 0, 1, 35, 392, 2352, 8820, 19404, 0, 1, 48, 720, 5760, 29700, 104544, 226512, 0, 1, 63, 1215, 12375, 81675, 382239, 1288287, 2760615, 0, 1, 80, 1925, 24200, 196625, 1145144, 5010005, 16359200, 34763300

Offset: 0

Peter Luschny, Dec 07 2024

			Array A(n, k) starts:
  [0] 0, 1,  3,   20,   175,    1764,    19404, ... A000891
  [1] 0, 1,  8,   75,   784,    8820,   104544, ... A145600
  [2] 0, 1, 15,  189,  2352,   29700,   382239, ... A145601
  [3] 0, 1, 24,  392,  5760,   81675,  1145144, ... A145602
  [4] 0, 1, 35,  720, 12375,  196625,  3006003, ... A145603
  [5] 0, 1, 48, 1215, 24200,  429429,  7154784, ...
  [6] 0, 1, 63, 1925, 44044,  869505, 15767024, ...
  [7] 0, 1, 80, 2904, 75712, 1656200, 32626944, ...
.
Seen as a triangle, T(n, k) = A(n-k, k). Compare the descending antidiagonals of A378061.
  [0] 0;
  [1] 0, 1;
  [2] 0, 1,  3;
  [3] 0, 1,  8,  20;
  [4] 0, 1, 15,  75,  175;
  [5] 0, 1, 24, 189,  784,  1764;
  [6] 0, 1, 35, 392, 2352,  8820,  19404;
  [7] 0, 1, 48, 720, 5760, 29700, 104544, 226512;

Rows: A000891, A145600, A145601, A145602, A145603.

Columns: A005563, A005565, A378061.

A := (n, k) -> ifelse(k = 0, 0, (n + 1)*binomial(2*k + n - 1, k - 1)^2/(2*k + n - 1)):
for n from 0 to 7 do seq(A(n, k), k = 0..7);

A[n_, k_] := If[k==0, 0, (n + 1)*Binomial[2*k + n - 1, k - 1]^2 / (2*k + n - 1)]; Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten (* Stefano Spezia, Dec 08 2024 *)

A067802 Triangle read by rows: T(n, k) = binomial(2n+1, n-k)^2(2k+1)/(2n+1).

Original entry on oeis.org

1, 3, 1, 20, 15, 1, 175, 189, 35, 1, 1764, 2352, 720, 63, 1, 19404, 29700, 12375, 1925, 99, 1, 226512, 382239, 196625, 44044, 4212, 143, 1, 2760615, 5010005, 3006003, 869505, 124215, 8085, 195, 1, 34763300, 66745536, 45048640, 15767024, 2998800, 299200, 14144, 255, 1

Offset: 0

Henry Bottomley, Feb 07 2002

			Triangle starts:
  [0]      1
  [1]      3,      1
  [2]     20,     15,      1
  [3]    175,    189,     35,     1
  [4]   1764,   2352,    720,    63,    1
  [5]  19404,  29700,  12375,  1925,   99,   1
  [6] 226512, 382239, 196625, 44044, 4212, 143, 1

First column is A000891.

Cf. A034869, A039599, A002894 (row sums).

T := (n, k) -> binomial(2*n+1, n-k)^2*(2*k+1)/(2*n+1):
seq(seq(T(n, k), k = 0..n), n = 0..8);  # Peter Luschny, Dec 07 2024

T(n, k) = A034869(2n+1, k) * A039599(n, k).

A138552 Returning walks of length 2n on the upper half of the square lattice, distinct under reflections about the y-axis.

Original entry on oeis.org

1, 2, 11, 90, 889, 9723, 113322, 1380522, 17382365, 224573349, 2962117366, 39741658047, 540862505806, 7450655906450, 103713126384420, 1456845308244810, 20627719676855685, 294136002612344145

Offset: 0

Andrew V. Sutherland, Mar 24 2008

nonn

Under reasonable assumptions, a(n)=E[X^{2n}] where the random variable X is the unitarized Frobenius trace X=a_p/sqrt(p) (as p varies) of a genus 2 curve whose Jacobian is isogenous to the product of two elliptic curves, exactly one of which has complex multiplication.

			a(2) = 11 because EEWW, EWEW, EWWE, EWNS, ENSW, ENWS, NEWS, NESW, NSEW, NSNS, NNSS are all the walks of length 4 on the upper half of the square lattice that are distinct under reflections about the y-axis.

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

Cf. A000108, A000891.

CoefficientList[Series[(3 Pi-2 Pi Sqrt[1-4x]-2EllipticE[16 x])/(8Pi x), {x, 0, 20}], x] (* Benedict W. J. Irwin, Jul 13 2016 *)

a(n) = (A000891(n) + A000108(n))/2.
G.f.: (3*Pi-2*Pi*sqrt(1-4*x)-2*EllipticE(16*x))/(8*Pi*x). - Benedict W. J. Irwin, Jul 13 2016
a(n) ~ 16^n*n^(-2)/Pi. - Ilya Gutkovskiy, Jul 13 2016
Recurrence: n*(n+1)^2*(3*n - 2)*a(n) = 2*n*(2*n - 1)*(15*n^2 - n - 4)*a(n-1) - 8*(2*n - 3)*(2*n - 1)^2*(3*n + 1)*a(n-2). - Vaclav Kotesovec, Jul 14 2016

A360444 a(n) is the number of ways for two nonintersecting, unordered pairs of shortest grid paths to cross over between two opposite corners in an n X n grid without intersecting opposite paths at their middle points.

Original entry on oeis.org

0, 0, 52, 4540, 742404, 103625004, 16451015760, 2693403573732, 463439672732740, 82516389937797244, 15153421065014201424, 2855078861978328905660, 550005952989718178915472, 108007620360200608500699120, 21569526154939330279935568704

Offset: 1

Janaka Rodrigo, Jul 15 2023

nonn

Alternatively, the number of different ways when two of four ants start at one corner of the grid and other two start at the opposite corner at the same time t and they all stop moving at time T (at which time each ant reaches the corner opposite from its starting corner) and at no time in the open interval (t,T) does any ant meet any other ant.

			In the 1 X 1 and 2 X 2 grids there is no possibility of this happening.
In a 3 X 3 grid, if a pair starting from the bottom left corner move along NNNEEE and EEENNN, the pair starting from the top right corner can move along WWSWSS and SSWSWW (this is only one of the nine options available for the second pair) so that they can cross over without meeting any other. There are 52 different ways to do this.

Cf. A362207, A358481, A357760, A357603, A000891.

A383823 a(n) is the number of secondary GL(4) invariants contructed from n+3 distinct four component vectors.

Original entry on oeis.org

1, 5, 105, 4116, 232848, 16818516, 1447482465, 142174944340, 15484613937936, 1832516612010448, 232187445047217296, 31148053701600494400

Offset: 1

Jaco van Zyl, May 11 2025

This number was evaluated by using the Molien-Weyl formula to compute the Hilbert series of the ring of invariants.

Robert de Mello Koch, Animik Ghosh, and Hendrik J. R. Van Zyl, Bosonic Fortuity in Vector Models, arXiv:2504.14181 [hep-th], 2025. See p. 9; Journal of High Energy Physics 06 (2025) 246.

Cf. A000891 (GL(2)), A382136 (GL(3)).

cp's OEIS Frontend

A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)*binomial(2*k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.

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A067802 Triangle read by rows: T(n, k) = binomial(2*n+1, n-k)^2*(2*k+1)/(2*n+1).

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A138552 Returning walks of length 2n on the upper half of the square lattice, distinct under reflections about the y-axis.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A360444 a(n) is the number of ways for two nonintersecting, unordered pairs of shortest grid paths to cross over between two opposite corners in an n X n grid without intersecting opposite paths at their middle points.

Views

Author

Keywords

Comments

Examples

Crossrefs

A383823 a(n) is the number of secondary GL(4) invariants contructed from n+3 distinct four component vectors.

Views

Author

Keywords

Comments

Links

Crossrefs

A378062 Array read by ascending antidiagonals: A(n, k) = (n + 1)binomial(2k + n - 1, k - 1)^2 / (2*k + n - 1) for k > 0, and A(n, 0) = 0.

A067802 Triangle read by rows: T(n, k) = binomial(2n+1, n-k)^2(2k+1)/(2n+1).