This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A079267 #77 Dec 10 2024 11:19:02
%S A079267 1,0,1,1,1,1,5,6,3,1,36,41,21,6,1,329,365,185,55,10,1,3655,3984,2010,
%T A079267 610,120,15,1,47844,51499,25914,7980,1645,231,21,1,721315,769159,
%U A079267 386407,120274,25585,3850,406,28,1,12310199,13031514,6539679,2052309,446544,70371,8106,666,36,1
%N A079267 d(n,s) = number of perfect matchings on {1, 2, ..., n} with s short pairs.
%C A079267 Read backwards, the n-th row of the triangle gives the Hilbert series of the variety of slopes determined by n points in the plane.
%C A079267 From _Paul Barry_, Nov 25 2009: (Start)
%C A079267 Reversal of coefficient array for the polynomials P(n,x) = Sum_{k=0..n} (C(n+k,2k)*(2k)!/(2^k*k!))*x^k*(1-x)^(n-k).
%C A079267 Note that P(n,x) = Sum_{k=0..n} A001498(n,k)*x^k*(1-x)^(n-k). (End)
%C A079267 Equivalent to the original definition: Triangle of fixed-point free involutions on [1..2n] (=A001147) by number of cycles with adjacent integers. - _Olivier Gérard_, Mar 23 2011
%C A079267 Conjecture: Asymptotically, the n-th row has a Poisson distribution with mean 1. - _David Callan_, Nov 11 2012
%C A079267 This is also the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_1 X P_2n (i.e., a path of length 2n) such that s such pairs are joined by an edge; equivalently the number of "s-domino" configurations in the game of memory played on a 1 X 2n rectangular array, see [Young]. - _Donovan Young_, Oct 23 2018
%D A079267 G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74.
%H A079267 Michael De Vlieger, <a href="/A079267/b079267.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened).
%H A079267 Marilena Barnabei, Niccolò Castronuovo, and Matteo Silimbani, <a href="https://arxiv.org/abs/2412.03449">Hertzsprung patterns on involutions</a>, arXiv:2412.03449 [math.CO], 2024. See p. 10.
%H A079267 Naiomi T. Cameron and Kendra Killpatrick, <a href="https://arxiv.org/abs/1902.09021">Statistics on Linear Chord Diagrams</a>, arXiv:1902.09021 [math.CO], 2019.
%H A079267 E. S. Krasko, I.N. Labutin, and A. V. Omelchenko, <a href="https://doi.org/10.1007/s10958-021-05350-1">Enumeration of labeled and unlabeled Hamiltonian cycles in complete k-partite graphs</a>, J. Math. Sci. 255 (2021) 71-87, eq (5).
%H A079267 G. Kreweras and Y. Poupard, <a href="/A000806/a000806.pdf">Sur les partitions en paires d'un ensemble fini totalement ordonné</a>, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74. (Annotated scanned copy)
%H A079267 Avichai Marmor, <a href="https://arxiv.org/abs/2307.09894">Schur-Positivity of Short Chords in Matchings</a>, arXiv:2307.09894 [math.CO], 2023.
%H A079267 Jeremy L. Martin, <a href="http://arxiv.org/abs/math/0302106">The slopes determined by n points in the plane</a>, arXiv:math/0302106 [math.AG], 2003-2006.
%H A079267 Jeremy L. Martin, <a href="http://dx.doi.org/10.1215/S0012-7094-05-13114-3">The slopes determined by n points in the plane</a>, Duke Math. J., Volume 131, Number 1 (2006), 119-165.
%H A079267 Donovan Young, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Young/young2.html">The Number of Domino Matchings in the Game of Memory</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.1.
%H A079267 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.
%H A079267 Donovan Young, <a href="https://arxiv.org/abs/2408.17232">Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams</a>, arXiv:2408.17232 [math.CO], 2024. See p. 18.
%F A079267 d(n, s) = (1/s!) * Sum_{h=s..n} (((-1)^(h-s)*(2*n-h)!/(2^(n-h)*(n-h)!*(h-s)!))).
%F A079267 E.g.f.: exp((x-1)*(1-sqrt(1-2*y)))/sqrt(1-2*y). - _Vladeta Jovovic_, Dec 15 2008
%e A079267 Triangle begins:
%e A079267 1
%e A079267 0 1
%e A079267 1 1 1
%e A079267 5 6 3 1
%e A079267 36 41 21 6 1
%e A079267 From _Paul Barry_, Nov 25 2009: (Start)
%e A079267 Production matrix begins
%e A079267 0, 1,
%e A079267 1, 1, 1,
%e A079267 4, 4, 2, 1,
%e A079267 18, 18, 9, 3, 1,
%e A079267 96, 96, 48, 16, 4, 1,
%e A079267 600, 600, 300, 100, 25, 5, 1,
%e A079267 4320, 4320, 2160, 720, 180, 36, 6, 1,
%e A079267 35280, 35280, 17640, 5880, 1470, 294, 49, 7, 1,
%e A079267 322560, 322560, 161280, 53760, 13440, 2688, 448, 64, 8, 1
%e A079267 Complete this by adding top row (1,0,0,0,...) and take inverse: we obtain
%e A079267 1,
%e A079267 0, 1,
%e A079267 -1, -1, 1,
%e A079267 -2, -2, -2, 1,
%e A079267 -3, -3, -3, -3, 1,
%e A079267 -4, -4, -4, -4, -4, 1,
%e A079267 -5, -5, -5, -5, -5, -5, 1,
%e A079267 -6, -6, -6, -6, -6, -6, -6, 1,
%e A079267 -7, -7, -7, -7, -7, -7, -7, -7, 1,
%e A079267 -8, -8, -8, -8, -8, -8, -8, -8, -8, 1 (End)
%e A079267 The 6 involutions with no fixed point on [1..6] with only one 2-cycle with adjacent integers are ((1, 2), (3, 5), (4, 6)), ((1, 3), (2, 4), (5, 6)), ((1, 3), (2, 6), (4, 5)), ((1, 5), (2, 3), (4, 6)), ((1, 5), (2, 6), (3, 4)), and ((1, 6), (2, 5), (3, 4)).
%p A079267 d := (n,s) -> 1/s! * sum('((-1)^(h-s)*(2*n-h)!/(2^(n-h)*(n-h)!*(h-s)!))','h'=s..n):
%p A079267 # alternative by _R. J. Mathar_, Aug 19 2022
%p A079267 A079267 := proc(n,k)
%p A079267 option remember ;
%p A079267 if n =0 and k =0 then
%p A079267 1;
%p A079267 elif k > n or k < 0 then
%p A079267 0;
%p A079267 else
%p A079267 procname(n-1,k-1)+(2*n-2-k)*procname(n-1,k)+(k+1)*procname(n-1,k+1) ;
%p A079267 end if;
%p A079267 end proc:
%p A079267 seq(seq( A079267(n,k),k=0..n),n=0..13) ;
%t A079267 nmax = 9; d[n_, s_] := (2^(s-n)*(2n-s)!* Hypergeometric1F1[s-n, s-2n, -2])/ (s!*(n-s)!); Flatten[ Table[d[n, s], {n, 0, nmax}, {s, 0, n}]] (* _Jean-François Alcover_, Oct 19 2011, after Maple *)
%o A079267 (PARI) {T(n, k) = 2^(k-n)*binomial(n,k)*hyperu(k-n, k-2*n, -2)};
%o A079267 for(n=0,10, for(k=0,n, print1(round(T(n,k)), ", "))) \\ _G. C. Greubel_, Apr 10 2019
%o A079267 (Sage) [[2^(k-n)*binomial(n,k)*hypergeometric_U(k-n,k-2*n,-2).simplify_hypergeometric() for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Apr 10 2019
%Y A079267 Columns are A278990, A000806, A006198, A006199, A006200.
%Y A079267 Row sums are A001147.
%Y A079267 d(2n,n) gives A365744.
%K A079267 easy,nice,nonn,tabl
%O A079267 0,7
%A A079267 Jeremy Martin (martin(AT)math.umn.edu), Feb 05 2003
%E A079267 Extra terms added by _Paul Barry_, Nov 25 2009