A055140 Triangle read by rows: T(n, k) = number of matchings of 2n people with partners (of either sex) such that exactly k couples are left together.

1, 0, 1, 2, 0, 1, 8, 6, 0, 1, 60, 32, 12, 0, 1, 544, 300, 80, 20, 0, 1, 6040, 3264, 900, 160, 30, 0, 1, 79008, 42280, 11424, 2100, 280, 42, 0, 1, 1190672, 632064, 169120, 30464, 4200, 448, 56, 0, 1, 20314880, 10716048, 2844288, 507360, 68544, 7560, 672, 72, 0, 1

Offset

0,4

Comments

T is an example of the group of matrices outlined in the table in A132382--the associated matrix for aC(1,1). The e.g.f. for the row polynomials is exp(x*t) * exp(-x) * (1-2*x)^(-1/2). T(n,k) = Binomial(n,k)* s(n-k) where s = A053871 with an e.g.f. of exp(-x) * (1-2*x)^(-1/2) which is the reciprocal of the e.g.f. of A055142. The row polynomials form an Appell sequence. Tom Copeland, Sep 10 2008

A231846 provides a refinement of this array. - Tom Copeland, Oct 12 2016

Links

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)

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.

Formula

T(n, k) = A053871(n-k)*binomial(n, k).

From Tom Copeland, Oct 12 2016: (Start)

E.g.f.: e^(xt) e^(-t) (1-2t)^(-1/2) = e^(p.(x)*t)(from my 2008 comment).

Row sums are A001147.

L = D = d/dx and R = x + d[log[e^(L)(1-2L)^(-1/2)]]/dL = x - 1 + 1/(1-2D) = x + 2D + (2D)^2 + (2D)^3 + ... are the lowering and raising operators, i.e., L p_n(x) = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x); e.g., L p_2(x) = D (2 + x^2) = 2 x = 2 p_1(x) and R P_2(x) = (x + 2D + 4D^2 + ...) (2 + x^2) = 2x + x^3 + 4x + 8 = 8 + 6x + x^3 = p_3(x).

Another generator is (1-2D)^(-1/2) e^(-D) x^n = (1-2D)^(-1/2) (x-1)^n = p_n(x). For example, (1-2D)^(-1/2)(x-1)^2 = (1 + D + 3 D^2/2 + ...) (x-1)^2 = (x-1)^2 + 2(x-1) + 3 = 2 + x^2 = p_2(x).

Umbral binomial convolution gives p_n(x) = (a. + x)^n = sum_{k = 0,..,n} C(n,k) a_(n-k) * x^k with (a.)^k = a_k = A053871(k).

The Appell sequence of umbral compositional inverses has the e.g.f. e^(xt) e^t (1-2t)^(1/2) associated with A055142. Cf. A231846 for a definition of umbral compositional inversion.

See A132382 and A133314 for more relations.

(End)

Example

Triangle T(n,k) starts:
     1;
     0,    1;
     2,    0,   1;
     8,    6,   0,   1;
    60,   32,  12,   0,  1;
   544,  300,  80,  20,  0, 1;
  6040, 3264, 900, 160, 30, 0, 1;
  ...

Maple

g[0] := 1: g[1] := 0: for n from 2 to 20 do g[n] := (2*(n-1))*(g[n-1]+g[n-2]) end do: T := proc (n, k) options operator, arrow; g[n-k]*binomial(n, k) end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form; Emeric Deutsch, Jan 24 2009

Mathematica

Table[(-1)^# HypergeometricPFQ[{1/2, -#}, {}, 2] Binomial[n, k] &[n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 10 2019, after Eric W. Weisstein at A053871 *)

Crossrefs

First column is A053871.

Row sums are A001147.

Cf. A055141, A055142.

Cf. A132382, A133314, A231846.

Sequence in context: A340556 A201637 A055141 * A335330 A191936 A327090

Adjacent sequences: A055137 A055138 A055139 * A055141 A055142 A055143

Keyword

nonn,tabl

Author

Christian G. Bower, May 09 2000

Status

approved