A334059 - OEIS

%I #24 Jan 07 2024 15:23:34

%S 1,0,1,1,2,0,5,8,2,0,36,49,19,1,0,329,414,180,22,0,0,3655,4398,1986,

%T 344,12,0,0,47844,55897,25722,5292,377,3,0,0,721315,825056,384366,

%U 87296,8746,246,0,0,0,12310199,13856570,6513530,1577350,192250,9436,90,0,0,0

%N 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.

%C 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].

%H Andrew Howroyd, <a href="/A334059/b334059.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%H Donovan Young, <a href="https://arxiv.org/abs/2004.06921">Linear k-Chord Diagrams</a>, arXiv:2004.06921 [math.CO], 2020.

%F 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)).

%e Triangle begins:

%e    1;

%e    0,  1;

%e    1,  2,  0;

%e    5,  8,  2, 0;

%e   36, 49, 19, 1  0;

%e   ...

%e 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.

%t 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}]

%o (PARI)

%o 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))}

%o { my(A=T(8)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, May 25 2020

%Y Row sums are A001147.

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

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

%K nonn,tabl

%O 0,5

%A _Donovan Young_, May 25 2020