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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.

%I A334059 #24 Jan 07 2024 15:23:34
%S A334059 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 A334059 344,12,0,0,47844,55897,25722,5292,377,3,0,0,721315,825056,384366,
%U A334059 87296,8746,246,0,0,0,12310199,13856570,6513530,1577350,192250,9436,90,0,0,0
%N 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.
%C A334059 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 A334059 Andrew Howroyd, <a href="/A334059/b334059.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A334059 Donovan Young, <a href="https://arxiv.org/abs/2004.06921">Linear k-Chord Diagrams</a>, arXiv:2004.06921 [math.CO], 2020.
%F A334059 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 A334059 Triangle begins:
%e A334059    1;
%e A334059    0,  1;
%e A334059    1,  2,  0;
%e A334059    5,  8,  2, 0;
%e A334059   36, 49, 19, 1  0;
%e A334059   ...
%e A334059 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 A334059 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 A334059 (PARI)
%o A334059 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 A334059 { my(A=T(8)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, May 25 2020
%Y A334059 Row sums are A001147.
%Y A334059 Column k=0 is A278990 (which is also column 0 of A079267).
%Y A334059 Cf. A079267, A334056, A334057, A334058, A325753.
%K A334059 nonn,tabl
%O A334059 0,5
%A A334059 _Donovan Young_, May 25 2020