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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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%I #48 Feb 15 2024 18:23:21

%S 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,

%T 30,0,1,79008,42280,11424,2100,280,42,0,1,1190672,632064,169120,30464,

%U 4200,448,56,0,1,20314880,10716048,2844288,507360,68544,7560,672,72,0,1

%N 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.

%C 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

%C A231846 provides a refinement of this array. - _Tom Copeland_, Oct 12 2016

%H Robert Israel, <a href="/A055140/b055140.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened)

%H Donovan Young, <a href="https://arxiv.org/abs/1905.13165">Generating Functions for Domino Matchings in the 2 * k Game of Memory</a>, arXiv:1905.13165 [math.CO], 2019. Also in <a href="https://www.emis.de/journals/JIS/VOL22/Young/young13.html">J. Int. Seq.</a>, Vol. 22 (2019), Article 19.8.7.

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

%F From _Tom Copeland_, Oct 12 2016: (Start)

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

%F Row sums are A001147.

%F 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).

%F 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).

%F 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).

%F 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.

%F See A132382 and A133314 for more relations.

%F (End)

%e Triangle T(n,k) starts:

%e 1;

%e 0, 1;

%e 2, 0, 1;

%e 8, 6, 0, 1;

%e 60, 32, 12, 0, 1;

%e 544, 300, 80, 20, 0, 1;

%e 6040, 3264, 900, 160, 30, 0, 1;

%e ...

%p 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

%t 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 *)

%Y First column is A053871.

%Y Row sums are A001147.

%Y Cf. A055141, A055142.

%Y Cf. A132382, A133314, A231846.

%K nonn,tabl

%O 0,4

%A _Christian G. Bower_, May 09 2000