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A084781 G.f. A(x) satisfies A(x) = 1 + x*(1+x+x^2)*A(x)^2. 0
1, 1, 3, 10, 35, 132, 519, 2105, 8746, 37033, 159229, 693343, 3051290, 13550083, 60642857, 273248824, 1238567263, 5643738611, 25837579578, 118785766683, 548182891007, 2538522337214, 11792272546723, 54936210525388 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(0)=1; for n > 0, a(n) = Sum_{j=n-3..n-1} Sum_{i=0..j} a(i)*a(j-i). - Mario Catalani (mario.catalani(AT)unito.it), Jun 19 2003

LINKS

Table of n, a(n) for n=0..23.

FORMULA

G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x+x^2+x^3 (continued fraction); equivalently g.f. C(x+x^2+x^3) where C(x) is the g.f. for the Catalan numbers (A000108). - Joerg Arndt, Mar 18 2011

a(n) = sum(m=1..n, (sum(k=m..n, ((sum(j=0..k, binomial(j,n-3*k+2*j) * binomial(k,j))) * binomial(-m+2*k-1,k-1))/k))*m). - Vladimir Kruchinin, May 28 2011

Recurrence: (n+1)*a(n) = 3*(n-1)*a(n-1) + (7*n-11)*a(n-2) + 12*(n-2)*a(n-3) + 2*(4*n-11)*a(n-4) + 2*(2*n-7)*a(n-5). - Vaclav Kotesovec, Sep 17 2013

a(n) ~ 1/sqrt(3)*sqrt(-(1350 + 66*sqrt(131)*sqrt(3))^(2/3) - 48 + 21*(1350 + 66*sqrt(131)*sqrt(3))^(1/3))/((1350 + 66*sqrt(131)*sqrt(3))^(1/6)) * (((190 + 6*sqrt(393))^(2/3) + 28 + 4*(190 + 6*sqrt(393))^(1/3))/(190 + 6*sqrt(393))^(1/3)/3)^n / (n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Sep 17 2013

MATHEMATICA

a[n_] := a[n] = Sum[Sum[a[i] a[j - i], {i, 0, j}], {j, n - 3, n - 1}]; a[0] = 1; Table[a[n], {n, 0, 30}]

Flatten[{1, Table[Sum[Sum[Sum[Binomial[j, n-3*k+2*j]*Binomial[k, j] *Binomial[-m+2*k-1, k-1]/k*m, {j, 0, k}], {k, m, n}], {m, 1, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Sep 17 2013 *)

PROG

(Maxima)

a(n):=sum((sum(((sum(binomial(j, n-3*k+2*j)*binomial(k, j), j, 0, k))*binomial(-m+2*k-1, k-1))/k, k, m, n))*m, m, 1, n); /* Vladimir Kruchinin, May 28 2011 */

CROSSREFS

Sequence in context: A296164 A151046 A221130 * A151047 A008984 A151048

Adjacent sequences:  A084778 A084779 A084780 * A084782 A084783 A084784

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 14 2003

STATUS

approved

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Last modified June 7 02:48 EDT 2020. Contains 334836 sequences. (Running on oeis4.)