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 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. 7
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified August 11 21:40 EDT 2024. Contains 375073 sequences. (Running on oeis4.)