A006251 Number of n-element posets which are unions of 2 chains. (Formerly M1216)

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

0,3

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

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

Mathematica

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

Prog

(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 */

Crossrefs

Sequence in context: A049143 A089404 A006123 * A049401 A294672 A239077

Adjacent sequences: A006248 A006249 A006250 * A006252 A006253 A006254

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Aug 21 2000

Status

approved