A341738 -id:A341738 - cp's OEIS frontend

A341741 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards: number of perfect matchings in the graph C_{2n} x C_k.

2, 8, 2, 14, 36, 2, 36, 50, 200, 2, 82, 272, 224, 1156, 2, 200, 722, 3108, 1058, 6728, 2, 478, 3108, 9922, 39952, 5054, 39204, 2, 1156, 10082, 90176, 155682, 537636, 24200, 228488, 2, 2786, 39952, 401998, 3113860, 2540032, 7379216, 115934, 1331716, 2

Offset: 1

Seiichi Manyama, Feb 18 2021

Dimer tilings of 2n x k toroidal grid.

			Square array begins:
  2,     8,    14,      36,       82,        200, ...
  2,    36,    50,     272,      722,       3108, ...
  2,   200,   224,    3108,     9922,      90176, ...
  2,  1156,  1058,   39952,   155682,    3113860, ...
  2,  6728,  5054,  537636,  2540032,  114557000, ...
  2, 39204, 24200, 7379216, 41934482, 4357599552, ...

Seiichi Manyama, Antidiagonals n = 1..50, flattened
P. W. Kasteleyn, The statistics of dimers on a lattice, I. the number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209-1225. See Eq. (25).
Index entries for sequences related to dominoes

Columns 1..12 give A007395, A162484(2*n), A231087, A220864(2*n), A231485, A232804(2*n), A230033, A253678(2*n), A281583, A281679(2*n), A308761, A309018(2*n).
Rows 1..6 give A162484, A220864, A232804, A253678, A281679, A309018.
T(n,2*n) gives A335586.
Cf. A341533, A341738, A341739.
Cf. A099390, A103997, A103999.

T(n,k) = A341533(n,k)/2 + A341738(n,k) + 2 * ((k+1) mod 2) * A341739(n,ceiling(k/2)).
T(n, 2k) = T(k, 2n).
If k is odd, T(n,k) = A341533(n,k) = 2*A341738(n,k).

New name from Andrey Zabolotskiy, Dec 26 2021

A341782 a(n) = sqrt( Product_{j=1..n} Product_{k=1..n-1} (4sin((2j-1)Pi/(2n))^2 + 4sin(2k*Pi/n)^2) ).

Original entry on oeis.org

1, 1, 2, 112, 2312, 1270016, 292820000, 1522266730496, 3772667519238272, 193509323594243571712, 5041011532336819845120512, 2610531939025273190037188509696

Offset: 0

Seiichi Manyama, Feb 19 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..62

Main diagonal of A341738.

Cf. A341478, A341535.

Table[Sqrt[Product[Product[(4*Sin[(2*j - 1)*Pi/(2*n)]^2 + 4*Sin[2*k*Pi/n]^2), {j, 1, n}], {k, 1, n - 1}]], {n, 0, 15}] // Round (* Vaclav Kotesovec, Mar 18 2023 *)

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, n-1, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin(2*k*Pi/n)^2))));

If n is odd, a(n) = A341535(n)/2.
If n is odd, a(n) = A341478(n).
a(n) ~ exp(2*G*n^2/Pi) / (2^(3/4) * (1 + (1 + (-1)^n)/sqrt(2))), where G is Catalan's constant A006752. - Vaclav Kotesovec, Mar 18 2023

A341739 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k} (4sin(aPi/n)^2 + 4sin((2b-1)Pi/(2k))^2).

Original entry on oeis.org

1, 1, 8, 1, 36, 49, 1, 200, 625, 288, 1, 1156, 12544, 9216, 1681, 1, 6728, 279841, 583200, 130321, 9800, 1, 39204, 6385729, 44408896, 24611521, 1822500, 57121, 1, 228488, 146410000, 3546167328, 6059221281, 1003520000, 25411681, 332928

Offset: 1

Seiichi Manyama, Feb 18 2021

			Square array begins:
     1,      1,        1,          1,             1, ...
     8,     36,      200,       1156,          6728, ...
    49,    625,    12544,     279841,       6385729, ...
   288,   9216,   583200,   44408896,    3546167328, ...
  1681, 130321, 24611521, 6059221281, 1612940640256, ...

Main diagonal gives A341478(n)^2.

Cf. A341533, A341738, A341741.

default(realprecision, 120);
T(n, k) = round(prod(a=1, n-1, prod(b=1, k, 4*sin(a*Pi/n)^2+4*sin((2*b-1)*Pi/(2*k))^2)));

cp's OEIS Frontend

A341741 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards: number of perfect matchings in the graph C_{2n} x C_k.

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A341782 a(n) = sqrt( Product_{j=1..n} Product_{k=1..n-1} (4sin((2j-1)Pi/(2n))^2 + 4sin(2k*Pi/n)^2) ).

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A341739 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k} (4sin(aPi/n)^2 + 4sin((2b-1)Pi/(2k))^2).

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cp's OEIS Frontend

A341741 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards: number of perfect matchings in the graph C_{2n} x C_k.

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Author

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Examples

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Crossrefs

Formula

Extensions

A341782 a(n) = sqrt( Product_{j=1..n} Product_{k=1..n-1} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin(2*k*Pi/n)^2) ).

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Author

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Mathematica

PARI

Formula

A341739 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k} (4*sin(a*Pi/n)^2 + 4*sin((2*b-1)*Pi/(2*k))^2).

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Author

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Programs

PARI

A341782 a(n) = sqrt( Product_{j=1..n} Product_{k=1..n-1} (4sin((2j-1)Pi/(2n))^2 + 4sin(2k*Pi/n)^2) ).

A341739 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k} (4sin(aPi/n)^2 + 4sin((2b-1)Pi/(2k))^2).