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A006251 Number of n-element posets which are unions of 2 chains.

%I A006251 M1216 #34 Jul 02 2025 16:01:54
%S A006251 1,1,2,4,10,26,75,225,711,2311,7725,26313,91141,319749,1134234,
%T A006251 4060128,14648614,53208998,194423568,714130372,2635256408,9764995800,
%U A006251 36320086418,135548135854,507434502474,1904982684106,7170113287574,27051804890638,102287657120454,387558371409606,1471212825012499,5594771416613721
%N A006251 Number of n-element posets which are unions of 2 chains.
%D A006251 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006251 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.45.
%H A006251 T. D. Noe, <a href="/A006251/b006251.txt">Table of n, a(n) for n=0..200</a>
%H A006251 R. P. Stanley (proposer), <a href="https://www.jstor.org/stable/2320567">Problem 6342</a>, Amer. Math. Monthly, 88 (1981), 294.
%H A006251 <a href="/index/Pos#posets">Index entries for sequences related to posets</a>
%F A006251 G.f.: 4/(2-2*x+sqrt(1-4*x)+sqrt(1-4*x^2)).
%F A006251 a(n) ~ (2-sqrt(3))*2^(2*n+3)/(6*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 13 2013
%F A006251 Recurrence: (n-1)*n*(n+2)*(n^2 - 8*n + 17)*a(n) = (n-1)*(9*n^4 - 74*n^3 + 159*n^2 + 36*n - 180)*a(n-1) - 2*(n-3)*(n-2)*(n-1)*(8*n^2 - 38*n + 15)*a(n-2) - 4*(n-1)*(14*n^4 - 194*n^3 + 999*n^2 - 2244*n + 1860)*a(n-3) + 8*(22*n^5 - 354*n^4 + 2259*n^3 - 7159*n^2 + 11307*n - 7125)*a(n-4) + 16*(n^5 - 37*n^4 + 392*n^3 - 1787*n^2 + 3681*n - 2775)*a(n-5) - 32*(2*n-9)*(6*n^4 - 100*n^3 + 612*n^2 - 1638*n + 1645)*a(n-6) + 64*(n-6)*(2*n-11)*(2*n-9)*(n^2 - 6*n + 10)*a(n-7). - _Vaclav Kotesovec_, Aug 13 2013
%t A006251 CoefficientList[Series[4/(2-2x+Sqrt[1-4x]+Sqrt[1-4x^2]), {x,0,40}], x] (* _Harvey P. Dale_, May 12 2011 *)
%o A006251 (PARI) x='x+O('x^44) /* that many terms */
%o A006251 gf=4/(2-2*x+sqrt(1-4*x)+sqrt(1-4*x^2));
%o A006251 Vec(gf) /* show terms */ /* _Joerg Arndt_, Apr 20 2011 */
%K A006251 nonn,nice,easy
%O A006251 0,3
%A A006251 _N. J. A. Sloane_
%E A006251 More terms from _James Sellers_, Aug 21 2000