OFFSET
0,4
COMMENTS
It appears that for n >= 1, a(n + 5) == a(n) (mod 5), a(n + 38*7) == a(n) (mod 7), a(n + 30*11) == a(n) (mod 11) and a(n + 288*17) == a(n) (mod 17). - Peter Bala, Apr 11 2022
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 0..201
O. Bodini, D. Gardy, and A. Jacquot, Asymptotics and random sampling for BCI and BCK lambda terms, Theor. Comput. Sci. 502: 227-238 (2013).
Katarzyna Grygiel, Pawel M. Idziak and Marek Zaionc, How big is BCI fragment of BCK logic, arXiv preprint arXiv:1112.0643 [cs.LO], 2011. (the authors of the paper incorrectly identified this sequence as A073950)
Pierre Lescanne, Quantitative aspects of linear and affine closed lambda term, arXiv:1702.03085 [cs.DM], 2017.
FORMULA
a(n) = 1 + a(n-1) + 2*Sum_{k=2..n-3} k*a(k) + Sum_{k=2..n-3} a(k)*a(n-1-k) for n>=2.
0 = 2*x^4*y' + (x-x^2)*y^2 - (1-x)^2*y + x^2, where y(x) is the g.f.
a(3*n+1) = Sum_{k=0..n-1} binomial(3*n,3*k+1)*A062980(k).
EXAMPLE
A(x) = x^2 + 2*x^3 + 3*x^4 + 9*x^5 + 30*x^6 + 81*x^7 + 242*x^8 + ...
MATHEMATICA
a[0] = a[1] = 0; a[n_] := a[n] = 1 + a[n - 1] + 2 Sum[ k a[k], {k, 2, n - 3}] + Sum[a[k] a[n - 1 - k], {k, 2, n - 3}]; Table[a@ n, {n, 0, 30}] (* Michael De Vlieger, Apr 02 2017 *)
PROG
(PARI)
seq(N) = {
my(a = vector(N));
for (n=2, N, my(s1 = sum(k=2, n-3, k*a[k]));
a[n] = 1 + a[n-1] + 2*s1 + sum(k=2, n-3, a[k]*a[n-1-k]));
concat(0, a);
};
seq(30)
\\ test: y = Ser(seq(201)); 0 == 2*x^4*y' + (x-x^2)*y^2 - (1-x)^2*y + x^2
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Apr 02 2017
EXTENSIONS
Name clarified by Pierre Lescanne, May 19 2017
STATUS
approved