A334059 Triangle read by rows: T(n,k) is the number of perfect matchings on {1, 2, ..., 2n} with k disjoint strings of adjacent short pairs.

1, 0, 1, 1, 2, 0, 5, 8, 2, 0, 36, 49, 19, 1, 0, 329, 414, 180, 22, 0, 0, 3655, 4398, 1986, 344, 12, 0, 0, 47844, 55897, 25722, 5292, 377, 3, 0, 0, 721315, 825056, 384366, 87296, 8746, 246, 0, 0, 0, 12310199, 13856570, 6513530, 1577350, 192250, 9436, 90, 0, 0, 0

Offset: 0

Author

Donovan Young, May 25 2020

Keywords

• nonn
• tabl

Comments

Number of configurations with k connected components (consisting of domino matchings) in the game of memory played on the path of length 2n, see [Young].

Examples

			Triangle begins:
   1;
   0,  1;
   1,  2,  0;
   5,  8,  2, 0;
  36, 49, 19, 1  0;
  ...
For n=2 and k=1 the configurations are (1,4),(2,3) (i.e. a single short pair) and (1,2),(3,4) (i.e. two adjacent short pairs); hence T(2,1) = 2.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
• Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020.

Crossrefs

Row sums are A001147.

Column k=0 is A278990 (which is also column 0 of A079267).

Cf. A079267, A334056, A334057, A334058, A325753.

Programs

Mathematica

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

PARI

T(n)={my(v=Vec(sum(j=0, n, (2*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(2*j+1) / (j! * 2^j * (1-(1-y)*x^2 + O(x*x^n))^(2*j+1))))); vector(#v, i, Vecrev(v[i], i))}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, May 25 2020

Formula

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