cp's OEIS Frontend

A088327 G.f.: exp(Sum_{k>=1} B(x^k)/k), where B(x) = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ... = (C(x)-1)/x and C is the g.f. for the Catalan numbers A000108.

%I A088327 #36 Aug 05 2025 21:32:22
%S A088327 1,1,3,8,25,77,256,854,2940,10229,36124,128745,463137,1677816,6118165,
%T A088327 22432778,82660369,305916561,1136621136,4238006039,15852603939,
%U A088327 59471304434,223704813807,843547443903,3188064830876,12074092672950,45816941923597,174173975322767
%N A088327 G.f.: exp(Sum_{k>=1} B(x^k)/k), where B(x) = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ... = (C(x)-1)/x and C is the g.f. for the Catalan numbers A000108.
%C A088327 a(n) is the number of forests of rooted plane binary trees (each node has outdegree = 0 or 2) where the trees have a total of n internal nodes. Cf. A222006. - _Geoffrey Critzer_, Feb 26 2013
%H A088327 Alois P. Heinz, <a href="/A088327/b088327.txt">Table of n, a(n) for n = 0..1000</a>
%F A088327 Euler transform of Catalan numbers (A000108). - _Franklin T. Adams-Watters_, Mar 01 2006
%F A088327 a(n) ~ c * 4^n / n^(3/2), where c = exp(Sum_{k>=1} (-2 + 4^k - 4^k*sqrt(1 - 4^(1-k)))/(2*k) ) / sqrt(Pi) = 1.60022306097485382475864802335610662545... - _Vaclav Kotesovec_, Mar 21 2021
%p A088327 with(numtheory):
%p A088327 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A088327        binomial(2*d, d)/(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
%p A088327     end:
%p A088327 seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 10 2012
%t A088327 With[{nn=35}, CoefficientList[Series[Product[1/(1-x^i)^CatalanNumber[i], {i,nn}], {x,0,nn}], x]] (* _Geoffrey Critzer_, Feb 26 2013 *)
%o A088327 (SageMath) # uses[EulerTransform from A166861]
%o A088327 b = EulerTransform(lambda n: binomial(2*n, n)/(n+1))
%o A088327 print([b(n) for n in range(30)]) # _Peter Luschny_, Nov 11 2020
%o A088327 (Magma)
%o A088327 m:=35;
%o A088327 f:= func< x | (&*[1/(1-x^j)^Catalan(j): j in [1..m+2]]) >;
%o A088327 R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( f(x) )); // _G. C. Greubel_, Dec 12 2022
%Y A088327 Row sums of A275431.
%Y A088327 Cf. A222006, A304787.
%K A088327 nonn
%O A088327 0,3
%A A088327 _N. J. A. Sloane_, Nov 06 2003