|
| |
|
|
A002003
|
|
2*Sum_{k=0..n-1} C(n-1,k)*C(n+k,k).
(Formerly M1857 N0735)
|
|
10
| |
|
|
0, 2, 8, 38, 192, 1002, 5336, 28814, 157184, 864146, 4780008, 26572086, 148321344, 830764794, 4666890936, 26283115038, 148348809216, 838944980514, 4752575891144, 26964373486406, 153196621856192, 871460014012682
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| a(n) is the number of order-preserving partial self maps of {1,...,n}. E.g. A002003 a(2)=8 because there are 8 order-preserving partial self maps of {1,2}: (1 2), (1 1), (2 2), (1 -), (2 -), (- 1), (- 2), (- -). Here for example (2 -) represents the partial map which maps 1 to 2 but does not include 2 in its domain. - James East (jameseastseq(AT)hotmail.com), Oct 25 2005
A002003(n+1) = A110110(2n+1) [From Tilman Neumann (Tilman.Neumann(AT)web.de), Feb 05 2009]
|
|
|
REFERENCES
| G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
FORMULA
| G.f.: ((1+x)/sqrt(1-6*x+x^2)-1)/2. E.g.f.: exp(3*x)*(2*BesselI(0, 2*sqrt(2)*x)+sqrt(2)*BesselI(1, 2*sqrt(2)*x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 28 2004
a(n) = 2 * JacobiP(n - 1, 0, 1, 3) = ((7*n+3)*LegendreP(n,3)-(n+1)*LegendreP(n+1,3))/(2*n) for n>0 [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Jul 12 2010]
Logarithmic derivative of A006318, the large Schroeder numbers. [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 25 2010]
|
|
|
MAPLE
| A064861 := proc(n, k) option remember; if n = 1 then 1; elif k = 0 then 0; else A064861(n, k-1)+(3/2-1/2*(-1)^(n+k))*A064861(n-1, k); fi; end; seq(A064861(i, i-1), i=1..40);
|
|
|
PROG
| (PARI) a(n)=if(n<1, 0, polcoeff(((1-x^2)/(1-x)^2+x*O(x^n))^n, n)) (from Michael Somos)
|
|
|
CROSSREFS
| Cf. A002002. Equals twice A047781.
a(n)=T(n, n-1), array T as in A064861, a(n)=T(n, n-2), array T as in A049600.
Cf. A006318. [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 25 2010]
Sequence in context: A155609 A199213 A123164 * A059423 A112109 A026939
Adjacent sequences: A002000 A002001 A002002 * A002004 A002005 A002006
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from Barbara Haas Margolius (b.margolius(AT)csuohio.edu), Oct 10, 2001
|
| |
|
|
