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

0,5

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].

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.

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)).

Example

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.

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}]

Prog

(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

Crossrefs

Row sums are A001147.

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

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

Sequence in context: A321205 A111352 A173343 * A133446 A011122 A329960

Adjacent sequences: A334056 A334057 A334058 * A334060 A334061 A334062

Keyword

nonn,tabl

Author

Donovan Young, May 25 2020

Status

approved