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A002003 a(n) = 2 * Sum_{k=0..n-1} C(n-1,k)*C(n+k,k).
(Formerly M1857 N0735)
12
0, 2, 8, 38, 192, 1002, 5336, 28814, 157184, 864146, 4780008, 26572086, 148321344, 830764794, 4666890936, 26283115038, 148348809216, 838944980514, 4752575891144, 26964373486406, 153196621856192, 871460014012682, 4962895187697048, 28292329581548718 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is the number of order-preserving partial self maps of {1,...,n}. For example, 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, Oct 25 2005

REFERENCES

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).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)

J. Brzozowski, M. Szykula, Large Aperiodic Semigroups, arXiv preprint arXiv:1401.0157, 2013.

Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

G. Rutledge and R. D. Douglass, Integral functions associated with certain binomial coefficient sums, Amer. Math. Monthly, 43 (1936), 27-32.

FORMULA

a(n) = 2*A047781(n).

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, Mar 28 2004

a(n) = T(n, n-1), array T as in A064861.

a(n) = T(n, n-2), array T as in A049600.

a(n+1) = A110110(2n+1). - Tilman Neumann, Feb 05 2009

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. - Mark van Hoeij, Jul 12 2010

Logarithmic derivative of A006318, the large Schroeder numbers. - Paul D. Hanna, Oct 25 2010

Recurrence: 4*(3*n^2-6*n+2)*a(n-1)-(n-2)*(2*n-1)*a(n-2)-n*(2*n-3)*a(n)=0. - Vaclav Kotesovec, Oct 04 2012

a(n) ~ (3+2*sqrt(2))^n/(2^(3/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 04 2012

Recurrence (an alternative): n*a(n) = (6-n)*a(n-6) + 2*(5*n-27)*a(n-5) + (84-15*n)*a(n-4) + 52*(3-n)*a(n-3) + 3*(2-5*n)*a(n-2) + 2*(5*n-3)*a(n-1), n>=7. - Fung Lam, Feb 05 2014

a(n) = Hyper2F1([-n, n], [1], -1) for n>0. - Peter Luschny, Aug 02 2014

a(n) = [x^n] ((1+x)/(1-x))^n for n > 0. - Seiichi Manyama, Jun 07 2018

EXAMPLE

G.f. = 2*x + 8*x^2 + 38*x^3 + 192*x^4 + 1002*x^5 + 5336*x^6 + 28814*x^7 + ...

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);

MATHEMATICA

Flatten[{0, Table[SeriesCoefficient[((1+x)/Sqrt[1-6*x+x^2]-1)/2, {x, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 04 2012 *)

a[ n_] := If[ n < 1, 0, Hypergeometric2F1[ n, -n, 1, -1]]; (* Michael Somos, Aug 24 2014 *)

PROG

(PARI) {a(n) = if( n<1, 0, polcoeff( ((1 - x^2) / (1 - x)^2 + x * O(x^n))^n, n))} /* Michael Somos, Sep 24 2003 */

CROSSREFS

Cf. A002002, A006318.

Sequence in context: A220542 A199213 A123164 * A264229 A059423 A112109

Adjacent sequences:  A002000 A002001 A002002 * A002004 A002005 A002006

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Barbara Haas Margolius (b.margolius(AT)csuohio.edu), Oct 10 2001

STATUS

approved

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Last modified January 25 07:37 EST 2020. Contains 331241 sequences. (Running on oeis4.)