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A213684
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Logarithmic derivative of A001002.
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7
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1, 5, 22, 105, 511, 2534, 12720, 64449, 328900, 1688115, 8705060, 45064110, 234054198, 1219053680, 6364813192, 33302104593, 174570695175, 916628799380, 4820160541350, 25381091113455, 133808636072595, 706211862466500, 3730964595817680, 19729042153581150
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OFFSET
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1,2
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COMMENTS
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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.
a(n) is the 2n-th term of the n-fold self-convolution of the Fibonacci numbers. - Alois P. Heinz, Feb 07 2021
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LINKS
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FORMULA
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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) = 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
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EXAMPLE
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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! +...
exp(L(x)) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 154*x^5 + 654*x^6 +...
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MAPLE
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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
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MATHEMATICA
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Table[n*Sum[Binomial[k+n-1, n]*Binomial[k, n-k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 20 2012 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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