A340427 -id:A340427 - cp's OEIS frontend

A340425 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of spanning trees of odd Aztec rectangle of order (n, k).

1, 4, 4, 16, 192, 16, 64, 8960, 8960, 64, 256, 417792, 4542720, 417792, 256, 1024, 19480576, 2280570880, 2280570880, 19480576, 1024, 4096, 908328960, 1143668117504, 12116689944576, 1143668117504, 908328960, 4096

Offset: 1

Seiichi Manyama, Jan 07 2021

			Square array begins:
    1,        4,            16,                64,                    256, ...
    4,      192,          8960,            417792,               19480576, ...
   16,     8960,       4542720,        2280570880,          1143668117504, ...
   64,   417792,    2280570880,    12116689944576,      64046643170770944, ...
  256, 19480576, 1143668117504, 64046643170770944, 3544863978266468352000, ...

Mihai Ciucu, Matchings and applications. Department of Mathematics, Indiana University, Bloomington, IN 47405. See Lecture 12.

Main diagonal gives A340352.

Cf. A340427.

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

# Using graphillion
from graphillion import GraphSet
def make_OD_nk(n, k):
    n += 1
    k += 1
    grids = []
    s = k * n
    for i in range(1, k * n, k):
        for j in range(1, k):
            a, b = i + j - 1, i + j
            c = s + a
            if i > 1:
                grids.extend([(c - k, a), (c - k, b)])
            if i < k * (n - 1) + 1:
                grids.extend([(c, a), (c, b)])
    return grids
def A340425(n, k):
    universe = make_OD_nk(n, k)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
print([A340425(j + 1, i - j + 1) for i in range(7) for j in range(i + 1)])

T(n,k) = T(k,n).
T(n,k) = 4^(2*n*k-n-k) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(a*Pi/(2*n))^2 * sin(b*Pi/(2*k))^2).
T(n,k) = 4^(2*n*k-n-k) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - cos(a*Pi/(2*n))^2 * cos(b*Pi/(2*k))^2).
T(n,k) = 4^(2*n*k-n-k) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(a*Pi/(2*n))^2 * cos(b*Pi/(2*k))^2).

A340428 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2nk) * Product_{a=1..n} Product_{b=1..k} (1 - sin(aPi/(2n+1))^2 * sin(bPi/(2k+1))^2).

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 61, 61, 1, 1, 547, 4961, 547, 1, 1, 4921, 432461, 432461, 4921, 1, 1, 44287, 38484961, 371647151, 38484961, 44287, 1, 1, 398581, 3445022461, 330435708793, 330435708793, 3445022461, 398581, 1

Offset: 0

Seiichi Manyama, Jan 07 2021

			Square array begins:
  1,    1,        1,            1,                1, ...
  1,    7,       61,          547,             4921, ...
  1,   61,     4961,       432461,         38484961, ...
  1,  547,   432461,    371647151,     330435708793, ...
  1, 4921, 38484961, 330435708793, 2952717950351617, ...

Main diagonal gives A340292.

Cf. A340427, A340430, A340432.

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

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

A340430 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2nk) * Product_{a=1..n} Product_{b=1..k} (1 - cos(aPi/(2n+1))^2 * cos(bPi/(2k+1))^2).

Original entry on oeis.org

1, 1, 1, 1, 15, 1, 1, 209, 209, 1, 1, 2911, 32625, 2911, 1, 1, 40545, 5015009, 5015009, 40545, 1, 1, 564719, 770100001, 8238791743, 770100001, 564719, 1, 1, 7865521, 118247646001, 13441754883649, 13441754883649, 118247646001, 7865521, 1

Offset: 0

Seiichi Manyama, Jan 07 2021

			Square array begins:
  1,     1,         1,              1,                  1, ...
  1,    15,       209,           2911,              40545, ...
  1,   209,     32625,        5015009,          770100001, ...
  1,  2911,   5015009,     8238791743,     13441754883649, ...
  1, 40545, 770100001, 13441754883649, 230629380093001665, ...

Main diagonal gives A340291.

Cf. A340427, A340428, A340432.

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

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

A340432 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2nk) * Product_{a=1..n} Product_{b=1..k} (1 - sin(aPi/(2n+1))^2 * cos(bPi/(2k+1))^2).

Original entry on oeis.org

1, 1, 1, 1, 13, 1, 1, 121, 181, 1, 1, 1093, 18281, 2521, 1, 1, 9841, 1690781, 2803921, 35113, 1, 1, 88573, 152963281, 2732887529, 430503601, 489061, 1, 1, 797161, 13755675781, 2555011015201, 4447515497881, 66102491401, 6811741, 1

Offset: 0

Seiichi Manyama, Jan 07 2021

			Square array begins:
  1,     1,         1,             1,                 1, ...
  1,    13,       121,          1093,              9841, ...
  1,   181,     18281,       1690781,         152963281, ...
  1,  2521,   2803921,    2732887529,     2555011015201, ...
  1, 35113, 430503601, 4447515497881, 43384923739812577, ...

Main diagonal gives A340295.

Cf. A340427, A340428, A340430.

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

cp's OEIS Frontend

A340425 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of spanning trees of odd Aztec rectangle of order (n, k).

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A340428 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2nk) * Product_{a=1..n} Product_{b=1..k} (1 - sin(aPi/(2n+1))^2 * sin(bPi/(2k+1))^2).

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A340430 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2nk) * Product_{a=1..n} Product_{b=1..k} (1 - cos(aPi/(2n+1))^2 * cos(bPi/(2k+1))^2).

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A340432 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2nk) * Product_{a=1..n} Product_{b=1..k} (1 - sin(aPi/(2n+1))^2 * cos(bPi/(2k+1))^2).

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cp's OEIS Frontend

A340425 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of spanning trees of odd Aztec rectangle of order (n, k).

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Author

Keywords

Examples

References

Crossrefs

Programs

PARI

Python

Formula

A340428 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2*n*k) * Product_{a=1..n} Product_{b=1..k} (1 - sin(a*Pi/(2*n+1))^2 * sin(b*Pi/(2*k+1))^2).

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A340430 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2*n*k) * Product_{a=1..n} Product_{b=1..k} (1 - cos(a*Pi/(2*n+1))^2 * cos(b*Pi/(2*k+1))^2).

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Formula

A340432 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2*n*k) * Product_{a=1..n} Product_{b=1..k} (1 - sin(a*Pi/(2*n+1))^2 * cos(b*Pi/(2*k+1))^2).

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Author

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Examples

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Programs

PARI

A340428 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2nk) * Product_{a=1..n} Product_{b=1..k} (1 - sin(aPi/(2n+1))^2 * sin(bPi/(2k+1))^2).

A340430 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2nk) * Product_{a=1..n} Product_{b=1..k} (1 - cos(aPi/(2n+1))^2 * cos(bPi/(2k+1))^2).

A340432 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 4^(2nk) * Product_{a=1..n} Product_{b=1..k} (1 - sin(aPi/(2n+1))^2 * cos(bPi/(2k+1))^2).