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A084782 G.f.: A(x) = 1 + x*A(x)^2/(1-x-x^2). 1
1, 1, 3, 11, 42, 168, 696, 2965, 12915, 57276, 257787, 1174597, 5407854, 25119663, 117579351, 554053049, 2626184688, 12513029640, 59898952650, 287931365692, 1389297316104, 6726449251539, 32668497856323, 159114598216251 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

FORMULA

a(0)=a(1)=1; for n>1, a(n) = Sum_{j=0..n} (Sum_{i=0..j} a(i)a(j-i))F(n-j), where F(n) are the Fibonacci numbers A000045. - Mario Catalani (mario.catalani(AT)unito.it), Jun 18 2003

a(n) = sum(sum(C(i,n-k-i)*C(k+i-1,k-1),i,ceiling((n-k)/2),n-k)*C(k),k,1,n), C(k) - Catalan numbers A000108. - Vladimir Kruchinin, Sep 15 2010

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

G.f.: 2/(sqrt((x^2+5*x-1)/(x^2+x-1))+1). - Vladimir Kruchinin, Oct 11 2011

Recurrence: (n+1)*a(n) = 3*(2*n-1)*a(n-1) - 3*(n-2)*a(n-2) - 3*(2*n-7) * a(n-3) - (n-5)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ 29^(1/4)*((5+sqrt(29))/2)^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

MATHEMATICA

CoefficientList[Series[2/(Sqrt[(x^2+5*x-1)/(x^2+x-1)]+1), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

PROG

(Maxima) a(n):=sum(sum(binomial(i, n-k-i)*binomial(k+i-1, k-1), i, ceiling((n-k)/2), n-k)*1/(k+1)*binomial(2*k, k), k, 1, n) /* Vladimir Kruchinin, Sep 15 2010 */

CROSSREFS

Sequence in context: A259858 A117641 A200030 * A149068 A151088 A149069

Adjacent sequences:  A084779 A084780 A084781 * A084783 A084784 A084785

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 14 2003

STATUS

approved

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Last modified November 13 13:15 EST 2018. Contains 317149 sequences. (Running on oeis4.)