A028484 -id:A028484 - cp's OEIS frontend

A334178 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 2^n * sqrt(Resultant(U_{2n}(x/2), T_{k}(ix/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

1, 1, 2, 1, 1, 4, 1, 3, 1, 8, 1, 4, 11, 1, 16, 1, 7, 19, 41, 1, 32, 1, 11, 71, 91, 153, 1, 64, 1, 18, 176, 769, 436, 571, 1, 128, 1, 29, 539, 2911, 8449, 2089, 2131, 1, 256, 1, 47, 1471, 17753, 48301, 93127, 10009, 7953, 1, 512, 1, 76, 4271, 79808, 603126, 801701, 1027207, 47956, 29681, 1, 1024

Offset: 0

Seiichi Manyama, Apr 17 2020

			Square array begins:
   1, 1,    1,     1,       1,        1,         1, ...
   2, 1,    3,     4,       7,       11,        18, ...
   4, 1,   11,    19,      71,      176,       539, ...
   8, 1,   41,    91,     769,     2911,     17753, ...
  16, 1,  153,   436,    8449,    48301,    603126, ...
  32, 1,  571,  2089,   93127,   801701,  20721019, ...
  64, 1, 2131, 10009, 1027207, 13307111, 714790675, ...

Rows 0..1 give A000012, A000032.
Columns 0..15 give A000079, A000012, A001835(n+1), A004253(n+1), A334135, A003729, A334179, A028478, A334180, A028480, A334181, A028482, A334182, A028484, A334183, A028486.
Cf. A103997.

T[n_, k_] := 2^n * Sqrt[Resultant[ChebyshevU[2*n, x/2], ChebyshevT[k, I*x/2], x]]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 04 2021 *)

{T(n, k) = sqrtint(4^n*polresultant(polchebyshev(2*n, 2, x/2), polchebyshev(k, 1, I*x/2)))}

T(n,2*k) = A103997(n,k) for k > 0.

A028486 Number of perfect matchings in graph C_{15} X P_{2n}.

Original entry on oeis.org

1, 1364, 6323504, 35269184041, 207171729355756, 1240837214254999769, 7491895591984935317759, 45390122553039546330628096, 275408624219475075609746445361, 1672150595320335623747680596071399, 10155382441518040205071335049138555724

Offset: 0

Per H. Lundow

nonn

For odd values of m the order of recurrence relation for the number of perfect matchings in C_{m} X P_{2n} graph does not exceed 2^floor(m/2). In general, this estimate is accurate, however the case m = 15 is an exception. This sequence obeys the recurrence relation of order 120. - Sergey Perepechko, Apr 28 2015

Per Hakan Lundow, "Computation of matching polynomials and the number of 1-factors in polygraphs", Research report, No 12, 1996, Department of Math., Umea University, Sweden.

Sergey Perepechko, Table of n, a(n) for n = 0..260
A. M. Karavaev, S. N. Perepechko, Dimer problem on cylinders: recurrences and generating functions, (in Russian), Matematicheskoe Modelirovanie, 2014, V.26, No.11, pp. 18-22.
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
Sergey Perepechko, Generating function for A028486

Cf. A028485, A028484.

{a(n) = sqrtint(4^n*polresultant(polchebyshev(2*n, 2, x/2), polchebyshev(15, 1, I*x/2)))} \\ Seiichi Manyama, Apr 17 2020

a(n) = 2^n * sqrt(Resultant(U_{2*n}(x/2), T_{15}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1). - Seiichi Manyama, Apr 17 2020

a(10) from Alois P. Heinz, Dec 10 2013

A340476 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4sin(aPi/(2n+1))^2 + 4cos(bPi/(2k+1))^2).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 19, 11, 1, 1, 91, 176, 29, 1, 1, 436, 2911, 1471, 76, 1, 1, 2089, 48301, 79808, 11989, 199, 1, 1, 10009, 801701, 4375897, 2091817, 97021, 521, 1, 1, 47956, 13307111, 240378643, 372713728, 53924597, 783511, 1364, 1

Offset: 0

Seiichi Manyama, Jan 09 2021

			Square array begins:
  1,  1,     1,       1,         1, ...
  1,  4,    19,      91,       436, ...
  1, 11,   176,    2911,     48301, ...
  1, 29,  1471,   79808,   4375897, ...
  1, 76, 11989, 2091817, 372713728, ...

Wikipedia, Chebyshev polynomials
Wikipedia, Resultant

Column k=0..1 give A000012, A002878.
Row n=0..7 give A000012, A004253(n+1), A003729, A028478, A028480, A028482, A028484, A028486.
Main diagonal gives A127606.
Cf. A187617, A340475.

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

{T(n, k) = sqrtint(4^k*polresultant(polchebyshev(2*n+1, 1, I*x/2), polchebyshev(2*k, 2, x/2)))}

T(n,k) = 2^k * sqrt(Resultant(T_{2*n+1}(i*x/2), U_{2*k}(x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

cp's OEIS Frontend

A334178 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 2^n * sqrt(Resultant(U_{2n}(x/2), T_{k}(ix/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A028486 Number of perfect matchings in graph C_{15} X P_{2n}.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A340476 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4sin(aPi/(2n+1))^2 + 4cos(bPi/(2k+1))^2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

cp's OEIS Frontend

A334178 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 2^n * sqrt(Resultant(U_{2*n}(x/2), T_{k}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A028486 Number of perfect matchings in graph C_{15} X P_{2n}.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A340476 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4*sin(a*Pi/(2*n+1))^2 + 4*cos(b*Pi/(2*k+1))^2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A334178 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = 2^n * sqrt(Resultant(U_{2n}(x/2), T_{k}(ix/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1).

A340476 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4sin(aPi/(2n+1))^2 + 4cos(bPi/(2k+1))^2).