A213684 Logarithmic derivative of A001002.

1, 5, 22, 105, 511, 2534, 12720, 64449, 328900, 1688115, 8705060, 45064110, 234054198, 1219053680, 6364813192, 33302104593, 174570695175, 916628799380, 4820160541350, 25381091113455, 133808636072595, 706211862466500, 3730964595817680, 19729042153581150

Offset

1,2

Comments

A001002(n) is the number of dissections of a convex (n+2)-gon into triangles and quadrilaterals by nonintersecting diagonals.

The g.f. of A001002 satisfies: G(x) = 1 + x*G(x)^2 + x^2*G(x)^3.

Central terms in A155161: a(n) = A155161(2*n,n). - Reinhard Zumkeller, Apr 17 2013

a(n) is the 2n-th term of the n-fold self-convolution of the Fibonacci numbers. - Alois P. Heinz, Feb 07 2021

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

D. Kruchinin and V. Kruchinin, A Generating Function for the Diagonal T2n,n in Triangles, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.

Formula

a(n) = n * Sum_{r=1..n} binomial(r+n-1,n) * binomial(r,n-r) / r.

L.g.f.: Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*(1+x)^n/n! = Sum_{n>=1} a(n)*x^n/n.

Recurrence: 75*(n-1)*n*a(n) = 5*(n-1)*(59*n-12)*a(n-1) + (559*n^2-1503*n+1100)* a(n-2) + 21*(3*n-8)*(3*n-7)*a(n-3). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 3^(3*n)/(2*5^(n-1/2)*sqrt(6*Pi*n)). - Vaclav Kotesovec, Oct 20 2012

a(n) = A037027(2*n-1,n-1). - Vladimir Kruchinin, Feb 28 2013

a(n) = Sum_{k=0..n} (-1)^floor((n-k)/2) [x^k] G(n,n,x/2), where G(n,a,x) denotes the n-th Gegenbauer polynomial; row sums of A298610. - Peter Luschny, Jan 26 2018

a(n) = [x^n] (1/(1-x-x^2))^n. - Alois P. Heinz, Feb 07 2021

Example

L.g.f.: L(x) = x + 5*x^2/2 + 22*x^3/3 + 105*x^4/4 + 511*x^5/5 +...
such that
L(x) = x*(1+x) + d/dx x^3*(1+x)^2/2! + d^2/dx^2 x^5*(1+x)^3/3! + d^3/dx^3 x^7*(1+x)^4/4! +...
The g.f. of A001002 begins:
exp(L(x)) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 154*x^5 + 654*x^6 +...

Maple

with(orthopoly): seq(add(i, i in [seq((-1)^iquo(n-k, 2)*coeff(G(n, n, x/2), x, k), k=0..n)]), n=1..24); # Peter Luschny, Jan 26 2018

Mathematica

Table[n*Sum[Binomial[k+n-1, n]*Binomial[k, n-k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 20 2012 *)

Prog

(PARI) {a(n)=n*sum(r=1, n, binomial(r+n-1, n)*binomial(r, n-r)/r)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A=(sum(m=1, n+1, Dx(m-1, x^(2*m-1)*(1+x)^m/m!)+x*O(x^n))); n*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(Haskell)
a213684 n = a155161 (2*n) n  -- Reinhard Zumkeller, Apr 17 2013

Crossrefs

Cf. A000045, A001002, A037027, A155161, A298610.

Sequence in context: A363546 A296163 A008485 * A373930 A082297 A267241

Adjacent sequences: A213681 A213682 A213683 * A213685 A213686 A213687

Keyword

nonn

Author

Paul D. Hanna, Jun 22 2012

Status

approved