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

1, 1, 2, 4, 10, 26, 75, 225, 711, 2311, 7725, 26313, 91141, 319749, 1134234, 4060128, 14648614, 53208998, 194423568, 714130372, 2635256408, 9764995800, 36320086418, 135548135854, 507434502474, 1904982684106, 7170113287574, 27051804890638, 102287657120454, 387558371409606, 1471212825012499, 5594771416613721

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.45.

Links

• T. D. Noe, Table of n, a(n) for n=0..200
• R. P. Stanley (proposer), Problem 6342, Amer. Math. Monthly, 88 (1981), 294.
• Index entries for sequences related to posets

Programs

Mathematica

CoefficientList[Series[4/(2-2x+Sqrt[1-4x]+Sqrt[1-4x^2]), {x,0,40}], x] (* Harvey P. Dale, May 12 2011 *)

PARI

x='x+O('x^44) /* that many terms */
gf=4/(2-2*x+sqrt(1-4*x)+sqrt(1-4*x^2));
Vec(gf) /* show terms */ /* Joerg Arndt, Apr 20 2011 */

Formula

G.f.: 4/(2-2*x+sqrt(1-4*x)+sqrt(1-4*x^2)).

a(n) ~ (2-sqrt(3))*2^(2*n+3)/(6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

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

Extensions

More terms from James Sellers, Aug 21 2000