|
%I #45 Mar 03 2024 14:45:45
%S 1,1,3,11,42,168,696,2965,12915,57276,257787,1174597,5407854,25119663,
%T 117579351,554053049,2626184688,12513029640,59898952650,287931365692,
%U 1389297316104,6726449251539,32668497856323,159114598216251
%N G.f.: A(x) = 1 + x*A(x)^2/(1-x-x^2).
%H Vincenzo Librandi, <a href="/A084782/b084782.txt">Table of n, a(n) for n = 0..200</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1910.00875">Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials</a>, arXiv:1910.00875 [math.CO], 2019.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 18.
%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F a(0) = a(1) = 1; for n>1, a(n) = Sum_{j=0..n-1} Fibonacci(n-j)*( Sum_{i=0..j} a(i)*a(j-i) ). - Mario Catalani (mario.catalani(AT)unito.it), Jun 18 2003
%F a(n) = Sum_{k=1..n} (Sum_{i=ceiling((n-k)/2)..n-k} binomial(i,n-k-i) *binomial(k+i-1,k-1) * C(k) ), C(k) - Catalan numbers A000108. - _Vladimir Kruchinin_, Sep 15 2010
%F 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
%F G.f.: 2/(sqrt((x^2+5*x-1)/(x^2+x-1)) + 1). - _Vladimir Kruchinin_, Oct 11 2011
%F 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
%F a(n) ~ 29^(1/4)*((5+sqrt(29))/2)^n/(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2012
%t CoefficientList[Series[2/(Sqrt[(x^2+5*x-1)/(x^2+x-1)]+1), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%o (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 */
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 2/(Sqrt((x^2+5*x-1)/(x^2+x-1)) + 1) )); // _G. C. Greubel_, Jun 07 2023
%o (SageMath)
%o @CachedFunction
%o def a(n): # a = A084782
%o if n<2: return 1
%o else: return sum( sum( a(k)*a(j-k) for k in range(j+1) )*fibonacci(n-j) for j in range(n) )
%o [a(n) for n in range(41)] # _G. C. Greubel_, Jun 07 2023
%Y Cf. A000045, A084781.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 14 2003
|
