

A334061


Triangle read by rows: T(n,k) is the number of set partitions of {1..4n} into n sets of 4 with k disjoint strings of adjacent sets, each being a contiguous set of elements


0



1, 0, 1, 31, 4, 0, 5474, 292, 9, 0, 2554091, 72318, 1206, 10, 0, 2502018819, 43707943, 438987, 2871, 5, 0, 4456194509950, 52717010017, 351487598, 1622954, 4355, 1, 0, 13077453070386914, 111615599664989, 528618296314, 1764575884, 4080889, 4385, 0, 0
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OFFSET

0,4


COMMENTS

Number of configurations with k connected components (consisting of polyomino matchings) in the generalized game of memory played on the path of length 4n, see [Young].


LINKS

Table of n, a(n) for n=0..35.
Donovan Young, Polyomino matchings in generalised games of memory and linear kchord diagrams, arXiv:2004.06921 [math.CO], 2020.


FORMULA

G.f.: Sum_{j>=0} (4*j)! * y^j * (1(1z)*y)^(4*j+1) / (j! * 24^j * (1(1z)*y^2)^(4*j+1)).


EXAMPLE

Triangle begins:
1;
0, 1;
31, 4, 0;
5474, 292, 9, 0;
2554091, 72318,1206, 10, 0;
...
For n=2 and k=1 the configurations are (1,6,7,8),(2,3,4,5), as well as (1,2,7,8),(3,4,5,6) and also (1,2,3,8),(4,5,6,7) (i.e. configurations with a single contiguous set) and (1,2,3,4),(5,6,7,8) (i.e. two adjacent contiguous sets); hence T(2,1) = 4.


MATHEMATICA

CoefficientList[Normal[Series[Sum[y^j*(4*j)!/24^j/j!*((1y*(1z))/(1y^2*(1z)))^(4*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}]


PROG

(PARI)
T(n)={my(v=Vec(sum(j=0, n, (4*j)! * x^j * (1(1y)*x + O(x*x^n))^(4*j+1) / (j! * 24^j * (1(1y)*x^2 + O(x*x^n))^(4*j+1))))); vector(#v, i, Vecrev(v[i], i))} { my(A=T(8)); for(n=1, #A, print(A[n])) }


CROSSREFS

Row sums are A025036.
Column k=0 is column 0 of A334057.
Cf. A079267, A334056, A334057, A334058, A334059, A334060, A325753.
Sequence in context: A107811 A109837 A128372 * A040942 A153072 A040943
Adjacent sequences: A334058 A334059 A334060 * A334062 A334063 A334064


KEYWORD

nonn,tabl


AUTHOR

Donovan Young, May 26 2020


STATUS

approved



