A028485 -id:A028485 - cp's OEIS frontend

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

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

A276053 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,2,4,6,8,10,...} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/3)).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 2, 2, 8, 7, 8, 4, 3, 26, 4, 13, 24, 24, 6, 66, 28, 8, 23, 62, 104, 8, 10, 158, 120, 64, 42, 148, 352, 80, 16, 19, 350, 416, 344, 16, 75, 334, 1052, 448, 160, 33, 756, 1252, 1440, 208, 32, 136, 726, 2860, 1936, 1024, 32, 61, 1578, 3448, 5176, 1440, 384, 244, 1534, 7312, 7056, 5072, 512, 64

Offset: 0

Emeric Deutsch, Aug 17 2016

The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the asymmetry degree of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).
Number of entries in row n is 1 + floor(n/3).
Sum of entries in row n is A028495(n).
T(n,0) = A276055(n)
Sum_{k>=0} k*T(n,k) = A276054(n).

			Row 4 is [4,2] because the compositions of 4 with parts in {1,2,4,6,8,...} are 4, 22, 211, 121, 112, and 1111, having asymmetry degrees 0, 0, 1, 0, 1, and 0, respectively.
Triangle starts:
  1;
  1;
  2;
  1,2;
  4,2;
  2,8;
  7,8,4.

S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.

Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239.
V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Cf. A028485, A276054, A276055.

G := (1-z^4)*(1+z-z^3)/(1-2*z^2-2*t*z^3-z^4+(3-2*t)*z^6+2*t*z^7-z^8): Gser := simplify(series(G, z = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form

Table[TakeWhile[BinCounts[#, {0, 1 + Floor[n/3], 1}], # != 0 &] &@ Map[Total, Map[Map[Boole[# >= 1] &, BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {_, a_, _} /; Nor[a == 1, EvenQ@ a]]], 1]]], {n, 0, 18}] // Flatten (* Michael De Vlieger, Aug 28 2016 *)

G.f.: G(t,z) = (1-z^4)*(1+z-z^3)/(1-2*z^2-2*t*z^3-z^4+(3-2*t)*z^6+2*t*z^7-z^8). In the more general situation of compositions into a[1]=1} z^(a[j]), we have G(t,z) = (1 + F(z))/(1 - F(z^2) - t*(F(z)^2 - F(z^2))). In particular, for t=0 we obtain Theorem 1.2 of the Hoggatt et al. reference.

cp's OEIS Frontend

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

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