A046741 -id:A046741 - cp's OEIS frontend

A265167 Number of n X 2 arrays containing 2 copies of 0..n-1 with no equal horizontal or vertical neighbors and new values introduced sequentially from 0.

0, 1, 2, 21, 186, 2113, 27856, 422481, 7241480, 138478561, 2923183474, 67520866405, 1694065383154, 45878853274945, 1333966056696224, 41446945223914337, 1370476678395567376, 48051281596087884289

Offset: 1

R. H. Hardin, Dec 03 2015

nonn

Column 2 of A265170.

a(n) is also the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that no such pair is joined by an edge; equivalently this is the number of "0-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young]. - Donovan Young, Oct 22 2018

			Some solutions for n=4
..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
..2..3....2..3....2..3....2..0....2..3....2..3....2..3....2..3....2..3....2..3
..0..1....3..2....0..2....1..3....0..1....3..0....3..1....1..2....3..2....0..2
..2..3....1..0....3..1....3..2....3..2....2..1....0..2....0..3....0..1....1..3

D. Young, The Number of Domino Matchings in the Game of Memory, Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.1.
Donovan Young, Generating Functions for Domino Matchings in the 2 * k Game of Memory, arXiv:1905.13165 [math.CO], 2019. Also in J. Int. Seq., Vol. 22 (2019), Article 19.8.7.

Cf. A265170, A046741.

a(n) = Sum_{k=0..n} (-1)^k*(2*n-2*k-1)!! * A046741(n,k) where and 0!! = (-1)!! = 1; proved by inclusion-exclusion, see [Young].

A102435 Triangle read by rows: T(n,k) is the number of k-matchings of the corona L'(n) of the ladder graph L(n)=P_2 X P_n. and the complete graph K(1); in other words, L'(n) is the graph constructed from L(n) by adding for each vertex v a new vertex v' and the edge vv'.

Original entry on oeis.org

1, 1, 3, 1, 1, 8, 16, 8, 1, 1, 13, 54, 87, 54, 13, 1, 1, 18, 117, 348, 501, 348, 117, 18, 1, 1, 23, 205, 914, 2210, 2966, 2210, 914, 205, 23, 1, 1, 28, 318, 1910, 6658, 13980, 17895, 13980, 6658, 1910, 318, 28, 1, 1, 33, 456, 3461, 15945, 46648, 88425, 109391, 88425, 46648, 15945, 3461, 456, 33, 1

Offset: 0

Emeric Deutsch, Jan 08 2005

Row n contains 2n+1 terms. Row sums yield A102436 T(n,k)=T(n,2n-k). The number of k-matchings of the ladder graph L(n)=P_2 X P_n is given in A046741.

			T(2,2)=16 because in the graph L'(2) with vertex set {A,B,C,D,a,b,c,d} and edge set {AB,BC,CD,AD,Aa,Bb,Cc,Dd} we have sixteen 2-matchings. Indeed, each of the edges Aa,Bb,Cc,Dd can be matched with five edges and each of the edges AB,BC,CD,AD can be matched with three edges. Thus we have (4*5+4*3)/2=16 matchings.
Triangle begins:
1;
1,3,1;
1,8,16,8,1;
1,13,54,87,54,13,1;

Cf. A102436, A046741.

P[0]:=1: P[1]:=1+3*t+t^2: P[2]:=1+8*t+16*t^2+8*t^3+t^4: for n from 3 to 8 do P[n]:=sort(expand((1+4*t+t^2)*P[n-1]+t*(1+t)^2*P[n-2]-t^3*P[n-3])) od: for n from 0 to 8 do seq(coeff(t*P[n],t^k),k=1..2*n+1) od; # yields sequence in triangular form

P[0]=1, P[1]=1+3t+t^2, P[2]=1+8t+16t^2+8t^3+t^4, P[n]=(1+4t+t^2)P[n-1]+t(1+t)^2*P[n-2]-t^3*P[n-3] for n>=3. G.f.= (1-tz)/[1-(1+4t+t^2)z-t(t+1)^2*z^2+t^3*z^3].

A123518 Number of dumbbells in all possible arrangements of dumbbells on a 2 X n rectangular array of compartments.

Original entry on oeis.org

1, 8, 38, 166, 671, 2602, 9792, 36068, 130697, 467556, 1655406, 5811290, 20255279, 70172502, 241839184, 829685064, 2835099649, 9653650752, 32768012102, 110913651342, 374469646511, 1261386990850, 4240037471152, 14225209349036

Offset: 1

Emeric Deutsch, Oct 16 2006

nonn

			a(2)=8 because in a 2 X 2 array of compartments, numbered clockwise starting from the NW one, we have 7 (=A030186(2)) possible arrangements of dumbbells: [ ], [14], [23], [12], [34], [14,23] and [12,34] (ij indicates a dumbbell placed in the compartments i and j); these contain altogether 8 dumbbells.

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
Index entries for linear recurrences with constant coefficients, signature (6,-7,-8,5,2,-1).

Cf. A030186, A046741.

a:=[1,8,38,166,671,2602];; for n in [7..30] do a[n]:=6*a[n-1] -7*a[n-2]-8*a[n-3]+5*a[n-4]+2*a[n-5]-a[n-6]; od; a; # G. C. Greubel, Oct 28 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(1+2*x-3*x^2+2*x^3)/(1-3*x-x^2+x^3)^2 )); // G. C. Greubel, Oct 28 2019

G:=z*(1+2*z-3*z^2+2*z^3)/(1-3*z-z^2+z^3)^2: Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=1..27);

LinearRecurrence[{6,-7,-8,5,2,-1}, {1,8,38,166,671,2602}, 30] (* G. C. Greubel, Oct 28 2019 *)

my(x='x+O('x^30)); Vec(x*(1+2*x-3*x^2+2*x^3)/(1-3*x-x^2+x^3)^2) \\ G. C. Greubel, Oct 28 2019

def A123518_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+2*x-3*x^2+2*x^3)/(1-3*x-x^2+x^3)^2 ).list()
a=A123518_list(30); a[1:] # G. C. Greubel, Oct 28 2019

a(n) = Sum_{k=0..n} k*A046741(n,k).

G.f.: x*(1 + 2*x - 3*x^2 + 2*x^3)/(1 - 3*x - x^2 + x^3)^2.

A325754 Triangle read by rows giving the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that exactly k such pairs are joined by a horizontal edge.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 7, 4, 4, 0, 43, 38, 21, 2, 1, 372, 360, 168, 36, 9, 0, 4027, 3972, 1818, 478, 93, 6, 1, 51871, 51444, 23760, 6640, 1260, 144, 16, 0, 773186, 768732, 358723, 103154, 20205, 2734, 278, 12, 1, 13083385, 13027060, 6129670, 1796740, 363595, 52900, 5650, 400, 25, 0

Offset: 0

Donovan Young, May 19 2019

This is the number of "k-horizontal-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young].

			The first few rows of T(n,k) are:
  1;
  1,  0;
  2,  0,  1;
  7,  4,  4,  0;
  43, 38, 21, 2, 1;
  ...
For n=2, let the vertex set of P_2 X P_2 be {A,B,C,D} and the edge set be {AB, AC, BD, CD}, where AB and CD are horizontal edges. For k=0, we may place the pairs on A, C and B, D or on A, D and B, C, hence T(2,0) = 2. If we place a pair on one of the horizontal edges we are forced to place the other pair on the remaining horizontal edge, hence T(2,1)=0 and T(2,2)=1.

D. Young, The Number of Domino Matchings in the Game of Memory, Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.1.
Donovan Young, Generating Functions for Domino Matchings in the 2 * k Game of Memory, arXiv:1905.13165 [math.CO], 2019. Also in J. Int. Seq., Vol. 22 (2019), Article 19.8.7.

Cf. A046741, A055140, A079267 A178523, A265167, A318243, A318244, A318267, A318268, A318269, A318270, A325753.

CoefficientList[Normal[Series[Sum[Factorial2[2*k-1]*y^k/(1-(1-z)*y)/(1+(1-z)*y)^(2*k+1), {k, 0, 20}], {y, 0, 20}]], {y, z}];

G.f.: Sum_{j>=0} (2*j-1)!! y^j/(1-(1-z)*y)/(1+(1-z)*y)^(2*j+1).

E.g.f.: exp((sqrt(1 - 2 y)-1) (1 - z))/sqrt(1 - 2 y) - exp((y - 2) (1 - z)) sqrt(Pi/2) sqrt(1 - z) (-erfi(sqrt(2) sqrt(1 - z)) + erfi(((1 + sqrt(1 - 2 y)) sqrt(1 - z))/sqrt(2))).

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A265167 Number of n X 2 arrays containing 2 copies of 0..n-1 with no equal horizontal or vertical neighbors and new values introduced sequentially from 0.

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A102435 Triangle read by rows: T(n,k) is the number of k-matchings of the corona L'(n) of the ladder graph L(n)=P_2 X P_n. and the complete graph K(1); in other words, L'(n) is the graph constructed from L(n) by adding for each vertex v a new vertex v' and the edge vv'.

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A123518 Number of dumbbells in all possible arrangements of dumbbells on a 2 X n rectangular array of compartments.

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A325754 Triangle read by rows giving the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that exactly k such pairs are joined by a horizontal edge.

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