

A035029


a(n) = Sum_{k=0..n} (k+1) * Sum_{j=0..n} 2^j*binomial(n,j)*binomial(nk,j).


4



1, 5, 26, 138, 743, 4043, 22180, 122468, 679757, 3789297, 21199998, 118973550, 669447123, 3775577367, 21336790152, 120795829128, 684962855705, 3889578815453, 22115533878178, 125892252068498, 717400693313471
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

Number of regions in all the dissections of a convex (n+3)gon by nonintersecting diagonals. a(1)=5 because in the three dissections of a square we have altogether five regions: one in the "nodiagonals" dissection and two in each of the dissections by one of the two diagonals of the square.  Emeric Deutsch, Dec 28 2003


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions
D. Merlini, R. Sprugnoli and M. C. Verri, Waiting patterns for a printer, FUN with algorithm'01, Isola d'Elba, 2001.


FORMULA

G.f.: (1z)^2/(8*z^2*sqrt(16*z+z^2))(1+z)/(8*z^2).  Emeric Deutsch, Dec 28 2003
a(n) = T(n+1, n+2), array T as in A049600.
Form an array with the m(n,1)=1 and m(1,n) = n*(n+1)/2 for n=1,2,3... The interior terms m(i,j) = m(i,j1) + m(i1,j1) + m(i1,j). The terms in the main diagonal = {a(n)}.  J. M. Bergot, Dec 01 2012
Dfinite with recurrence: (n+2)*a(n) + (7*n8)*a(n1) + (7*n8)*a(n2) + (n+2)*a(n3) = 0.  R. J. Mathar, Dec 03 2012
a(n) ~ (3+2*sqrt(2))^(n+3/2) / (2^(9/4)*sqrt(Pi*n)).  Vaclav Kotesovec, Feb 12 2014
Conjecture: a(n) = 2F1(n,n+3; 1; 1), which would satisfy above recurrence.  Benedict W. J. Irwin, Oct 14 2016


MATHEMATICA

CoefficientList[Series[(1x)^2/(8*x^2*Sqrt[16*x+x^2])(1+x)/(8*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)


CROSSREFS

Equals (1/4) [A002002(n+1)  A002002(n)].
Cf. A001003.
Sequence in context: A288785 A161731 A049607 * A081569 A005573 A081911
Adjacent sequences: A035026 A035027 A035028 * A035030 A035031 A035032


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


STATUS

approved



