A055141 -id:A055141 - cp's OEIS frontend

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

Christian G. Bower, May 09 2000

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

			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;
  ...

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.

First column is A053871.
Row sums are A001147.
Cf. A055141, A055142.
Cf. A132382, A133314, A231846.

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

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 *)

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)

A055142 Expansion of e.g.f.: exp(x)*sqrt(1-2x).

Original entry on oeis.org

1, 0, -2, -8, -36, -224, -1880, -19872, -251888, -3712256, -62286624, -1171487360, -24402416192, -557542291968, -13861636770176, -372514645389824, -10759590258589440, -332386419622387712

Offset: 0

Christian G. Bower, May 09 2000

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001147, A053871.

Column 0 of triangle A055141.

CoefficientList[Series[E^x*Sqrt[1-2*x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 29 2014 *)

def A055142(n):
    @CachedFunction
    def h(n):
        return (-1)^n*2*(n-2)*abs(h(n-1)+h(n-2)) if n>1 else 1
    return -(-1)^(n+1)*h(n+1)
[A055142(n) for n in (0..17)]  # Peter Luschny, Nov 02 2012

a(n) ~ -2^(n-1/2) * n^(n-1) / exp(n-1/2). - Vaclav Kotesovec, Mar 29 2014

D-finite with recurrence: a(n) +2*(-n+1)*a(n-1) +2*(n-1)*a(n-2)=0. - R. J. Mathar, Jan 22 2020

cp's OEIS Frontend

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.

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