OFFSET
1,2
COMMENTS
First column of triangle A100326. - Paul D. Hanna, Nov 16 2004
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..200
M. Bicknell and V. E. Hoggatt, Jr., Sequences of matrix inverses from Pascal, Catalan and related convolution arrays, Fib. Quart., 14 (1976), 224-232.
L. Carlitz, Enumeration of two-line arrays, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 416
R. C. Read, On the enumeration of a class of plane multigraphs, Aequat. Math. 31 (1986) no 1, 47-63.
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.
FORMULA
For formula see Read reference.
D-finite with recurrence a(n) = ( (324*n^2-708*n+360)*a(n-1) - (371*n^2-1831*n+2250)*a(n-2) + (20*n^2-130*n+210)*a(n-3) )/(16*n*(2*n-1)) for n>2, with a(0)=0, a(1)=1, a(2)=3. - Paul D. Hanna, Nov 16 2004
G.f. satisfies: A(x) = x*(1+A(x))/(1-A(x))^2 where A(0)=0. G.f. satisfies: (1+A(x))/(1-A(x)) = 2*G003168(x)-1, where G003168 is the g.f. of A003168. - Paul D. Hanna, Nov 16 2004
a(n) = (1/n)*Sum_{i=0..n-1} binomial(n,i)*binomial(3*n-i-2,n-i-1). - Vladeta Jovovic, Sep 13 2006
Appears to be (1/n)*Jacobi_P(n-1,1,n-1,3). If so then a(n) = (1/(2*n-1))*Sum_{k = 0..n-1} binomial(n-1,k)*binomial(2*n+k-1,k+1) = (1/n)*Sum_{k = 0..n} binomial(n,k)*binomial(2*n-2,n+k-1)*2^k. [Added Jun 11 2023: these are correct, and can be proved using the WZ algorithm.] - Peter Bala, Aug 01 2012
a(n) ~ sqrt(33/sqrt(17)-7) * ((71+17*sqrt(17))/16)^n / (4*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 09 2013
The o.g.f. A(x) = 1 + 3*x + 14*x^2 + ... taken with offset 0, satisfies 1 + x*A'(x)/A(x) = 1 + 3*x + 19*x^2 + 138*x^3 + ..., the o.g.f. for A156894. - Peter Bala, Oct 05 2015
From Peter Bala, Jun 11 2023: (Start)
a(n) = (1/n)*Sum_{k = 0..n-1} binomial(n,k+1)*binomial(2*n+k-1,k) (Carlitz, equation 3.19).
4*n*(17*n - 29)*(2*n - 1)*a(n) = (1207*n^3 - 4473*n^2 + 5258*n - 1920)*a(n-1) - 2*(2*n - 5)*(17*n - 12)*(n - 2)*a(n-2) with a(1) = 1 and a(2) = 3. (End)
MAPLE
a[0]:=0:a[1]:=1:a[2]:=3:for n from 3 to 30 do a[n]:=((324*n^2-708*n+360)*a[n-1] -(371*n^2-1831*n+2250)*a[n-2]+(20*n^2-130*n+210)*a[n-3])/(16*n*(2*n-1)) od:seq(a[n], n=1..25); # Emeric Deutsch, Jan 31 2005
MATHEMATICA
lim = 21; t[0, 0] = 1; t[n_, 0] := t[n, 0] = Sum[(k + 1)*t[n - 1, k], {k, 0, n - 1}]; t[n_, k_] := t[n, k] = Sum[t[j + 1, 0]*t[n - j - 1, k - 1], {j, 0, n - k}]; Table[ t[n, 0], {n, lim}] (* Jean-François Alcover, Sep 20 2011, after Paul D. Hanna's comment *)
PROG
(PARI) {a(n)=if(n==0, 0, if(n==1, 1, if(n==2, 3, ( (324*n^2-708*n+360)*a(n-1) -(371*n^2-1831*n+2250)*a(n-2)+(20*n^2-130*n+210)*a(n-3))/(16*n*(2*n-1)) )))} \\ Paul D. Hanna, Nov 16 2004
(PARI) {a(n)=local(A=x+x*O(x^n)); if(n==1, 1, for(i=1, n, A=x*(1+A)/(1-A)^2); polcoeff(A, n))}
(PARI) seq(n)=Vec(serreverse(x*(1 - x)^2/(1 + x) + O(x*x^n))) \\ Andrew Howroyd, Mar 07 2023
(Haskell)
a003169 = flip a100326 0 -- Reinhard Zumkeller, Nov 21 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, Jan 31 2005
STATUS
approved