A325753 -id:A325753 - cp's OEIS frontend

A334056 Triangle read by rows: T(n,k) is the number of configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 3n.

1, 0, 1, 7, 2, 1, 219, 53, 7, 1, 12861, 2296, 226, 16, 1, 1215794, 171785, 13080, 710, 30, 1, 169509845, 19796274, 1228655, 53740, 1835, 50, 1, 32774737463, 3260279603, 170725639, 6250755, 178325, 4137, 77, 1, 8400108766161, 727564783392, 32944247308, 1036855344, 25359670, 507584, 8428, 112, 1

Offset: 0

Donovan Young, Apr 15 2020

In this generalized game of memory n indistinguishable triples of matched cards are placed on the vertices of the path of length 3n. A polyomino is a triple on three adjacent vertices. For dominoes in ordinary memory on the path of length 2n, see A079267.

T(n,k) is the number of set partitions of {1..3n} into n sets of 3 with k of the sets being a contiguous set of elements. - Andrew Howroyd, Apr 16 2020

			The first few rows of T(n,k) are:
      1;
      0,    1;
      7,    2,   1;
    219,   53,   7,  1;
  12861, 2296, 226, 16, 1;
  ...
For n=2 and k=1 the polyomino must start either on the second vertex of the path, or the third, otherwise the remaining triple will also form a polyomino; thus T(2,1) = 2.

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. See also J. Int. Seq., Vol. 23 (2020), Article 20.9.1.

Row sums are A025035.

Cf. A079267, A334057, A334058, A325753.

CoefficientList[Normal[Series[Sum[y^j*(3*j)!/6^j/j!/(1+y*(1-z))^(3*j+1),{j,0,20}],{y,0,20}]],{y,z}]

T(n,k)={sum(j=0, n-k, (-1)^(n-j-k)*(n+2*j)!/(6^j*j!*(n-j-k)!*k!))} \\ Andrew Howroyd, Apr 16 2020

G.f.: Sum_{j>=0} (3*j)! * y^j / (j! * 6^j * (1+(1-z)*y)^(3*j+1)).

T(n,k) = Sum_{j=0..n-k} (-1)^(n-j-k)*(n+2*j)!/(6^j*j!*(n-j-k)!*k!). - Andrew Howroyd, Apr 16 2020

A334057 Triangle read by rows: T(n,k) is the number of configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 4n.

Original entry on oeis.org

1, 0, 1, 31, 3, 1, 5474, 288, 12, 1, 2554091, 72026, 1476, 31, 1, 2502018819, 43635625, 508610, 5505, 65, 1, 4456194509950, 52673302074, 394246455, 2559565, 16710, 120, 1, 13077453070386914, 111562882654972, 580589062179, 2504572910, 10288390, 43806, 203, 1

Offset: 0

Donovan Young, Apr 15 2020

In this generalized game of memory n indistinguishable quadruples of matched cards are placed on the vertices of the path of length 4n. A polyomino is a quadruple on four adjacent vertices.

T(n,k) is the number of set partitions of {1..4n} into n sets of 4 with k of the sets being a contiguous set of elements. - Andrew Howroyd, Apr 16 2020

			The first few rows of T(n,k) are:
        1;
        0,     1;
       31,     3,    1;
     5474,   288,   12,  1;
  2554091, 72026, 1476, 31, 1;
  ...
For n=2 and k=1 the polyomino must start either on the second vertex of the path, the third, or the fourth, otherwise the remaining quadruple will also form a polyomino; thus T(2,1) = 3.

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.

Row sums are A025036.

Cf. A079267, A334056, A334058, A325753.

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

T(n,k)={sum(j=0, n-k, (-1)^(n-j-k)*(n+3*j)!/(24^j*j!*(n-j-k)!*k!))} \\ Andrew Howroyd, Apr 16 2020

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

T(n,k) = Sum_{j=0..n-k} (-1)^(n-j-k)*(n+3*j)!/(24^j*j!*(n-j-k)!*k!). - Andrew Howroyd, Apr 16 2020

A334058 Triangle read by rows: T(n,k) is the number of configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 5n.

Original entry on oeis.org

1, 0, 1, 121, 4, 1, 124760, 1347, 18, 1, 486854621, 2001548, 8154, 52, 1, 5184423824705, 10231953233, 17045774, 35542, 121, 1, 123243726413573515, 134835947255262, 112619668659, 102416812, 124881, 246, 1, 5717986519188343198259, 3821094862609800013, 1820735766620673, 863827126967, 486979381, 375627, 455, 1

Offset: 0

Donovan Young, Apr 15 2020

In this generalized game of memory n indistinguishable quintuples of matched cards are placed on the vertices of the path of length 5n. A polyomino is a quintuple on five adjacent vertices.

T(n,k) is the number of set partitions of {1..5n} into n sets of 5 with k of the sets being a contiguous set of elements. - Andrew Howroyd, Apr 16 2020

			The first few rows of T(n,k) are:
          1;
          0,       1;
        121,       4,    1;
     124760,    1347,   18,  1;
  486854621, 2001548, 8154, 52, 1;
  ...
For n=2 and k=1 the polyomino must start either on the second, third, fourth, or fifth vertex of the path, otherwise the remaining quintuple will also form a polyomino; thus T(2,1) = 4.

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.

Row sums are A025037.

Cf. A079267, A334056, A334057, A325753.

CoefficientList[Normal[Series[Sum[y^j*(5*j)!/120^j/j!/(1+y*(1-z))^(5*j+1),{j,0,20}],{y,0,20}]],{y,z}]

T(n,k)={sum(j=0, n-k, (-1)^(n-j-k)*(n+4*j)!/(120^j*j!*(n-j-k)!*k!))} \\ Andrew Howroyd, Apr 16 2020

G.f.: Sum_{j>=0} (5*j)! * y^j / (j! * 120^j * (1+(1-z)*y)^(5*j+1)).

T(n,k) = Sum_{j=0..n-k} (-1)^(n-j-k)*(n+4*j)!/(120^j*j!*(n-j-k)!*k!). - Andrew Howroyd, Apr 16 2020

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.

Original entry on oeis.org

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

Donovan Young, May 25 2020

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].

			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.

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.

Row sums are A001147.
Column k=0 is A278990 (which is also column 0 of A079267).
Cf. A079267, A334056, A334057, A334058, A325753.

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}]

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

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)).

A334060 Triangle read by rows: T(n,k) is the number of set partitions of {1..3n} into n sets of 3 with k disjoint strings of adjacent sets, each being a contiguous set of elements.

Original entry on oeis.org

1, 0, 1, 7, 3, 0, 219, 56, 5, 0, 12861, 2352, 183, 4, 0, 1215794, 174137, 11145, 323, 1, 0, 169509845, 19970411, 1078977, 30833, 334, 0, 0, 32774737463, 3280250014, 153076174, 4056764, 55379, 206, 0, 0, 8400108766161, 730845033406, 29989041076, 727278456, 10341101, 67730, 70, 0, 0

Offset: 0

Donovan Young, May 26 2020

Number of configurations with k connected components (consisting of polyomino matchings) in the generalized game of memory played on the path of length 3n, see [Young].

			Triangle begins:
      1;
      0,    1;
      7,    3,   0;
    219,   56,   5, 0;
  12861, 2352, 183, 4, 0;
  ...
For n=2 and k=1 the configurations are (1,5,6),(2,3,4) and (1,2,6),(3,4,5) (i.e. configurations with a single contiguous set) and (1,2,3),(4,5,6) (i.e. two adjacent contiguous sets); hence T(2,1) = 3.

Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020.

Row sums are A025035.
Column k=0 is column 0 of A334056.
Cf. A079267, A334056, A334057, A334058, A334059, A325753.

CoefficientList[Normal[Series[Sum[y^j*(3*j)!/6^j/j!*((1-y*(1-z))/(1-y^2*(1-z)))^(3*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}]

T(n)={my(v=Vec(sum(j=0, n, (3*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(3*j+1) / (j! * 6^j * (1-(1-y)*x^2 + O(x*x^n))^(3*j+1))))); vector(#v, i, Vecrev(v[i], i))}
{ my(A=T(8)); for(n=1, #A, print(A[n])) }

G.f.: Sum_{j>=0} (3*j)! * y^j * (1-(1-z)*y)^(3*j+1) / (j! * 6^j * (1-(1-z)*y^2)^(3*j+1)).

A325754 Triangle read by rows giving the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that exactly k such pairs are joined by a horizontal edge.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 7, 4, 4, 0, 43, 38, 21, 2, 1, 372, 360, 168, 36, 9, 0, 4027, 3972, 1818, 478, 93, 6, 1, 51871, 51444, 23760, 6640, 1260, 144, 16, 0, 773186, 768732, 358723, 103154, 20205, 2734, 278, 12, 1, 13083385, 13027060, 6129670, 1796740, 363595, 52900, 5650, 400, 25, 0

Offset: 0

Donovan Young, May 19 2019

This is the number of "k-horizontal-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young].

			The first few rows of T(n,k) are:
  1;
  1,  0;
  2,  0,  1;
  7,  4,  4,  0;
  43, 38, 21, 2, 1;
  ...
For n=2, let the vertex set of P_2 X P_2 be {A,B,C,D} and the edge set be {AB, AC, BD, CD}, where AB and CD are horizontal edges. For k=0, we may place the pairs on A, C and B, D or on A, D and B, C, hence T(2,0) = 2. If we place a pair on one of the horizontal edges we are forced to place the other pair on the remaining horizontal edge, hence T(2,1)=0 and T(2,2)=1.

D. Young, The Number of Domino Matchings in the Game of Memory, Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.1.
Donovan Young, Generating Functions for Domino Matchings in the 2 * k Game of Memory, arXiv:1905.13165 [math.CO], 2019. Also in J. Int. Seq., Vol. 22 (2019), Article 19.8.7.

Cf. A046741, A055140, A079267 A178523, A265167, A318243, A318244, A318267, A318268, A318269, A318270, A325753.

CoefficientList[Normal[Series[Sum[Factorial2[2*k-1]*y^k/(1-(1-z)*y)/(1+(1-z)*y)^(2*k+1), {k, 0, 20}], {y, 0, 20}]], {y, z}];

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

E.g.f.: exp((sqrt(1 - 2 y)-1) (1 - z))/sqrt(1 - 2 y) - exp((y - 2) (1 - z)) sqrt(Pi/2) sqrt(1 - z) (-erfi(sqrt(2) sqrt(1 - z)) + erfi(((1 + sqrt(1 - 2 y)) sqrt(1 - z))/sqrt(2))).

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.

Original entry on oeis.org

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

Offset: 0

Donovan Young, May 26 2020

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].

			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.

Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020.

Row sums are A025036.
Column k=0 is column 0 of A334057.
Cf. A079267, A334056, A334057, A334058, A334059, A334060, A325753.

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

T(n)={my(v=Vec(sum(j=0, n, (4*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(4*j+1) / (j! * 24^j * (1-(1-y)*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])) }

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

cp's OEIS Frontend

A334056 Triangle read by rows: T(n,k) is the number of configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 3n.

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A334057 Triangle read by rows: T(n,k) is the number of configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 4n.

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A334058 Triangle read by rows: T(n,k) is the number of configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 5n.

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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.

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A334060 Triangle read by rows: T(n,k) is the number of set partitions of {1..3n} into n sets of 3 with k disjoint strings of adjacent sets, each being a contiguous set of elements.

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A325754 Triangle read by rows giving the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that exactly k such pairs are joined by a horizontal edge.

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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.

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