OFFSET
1,3
COMMENTS
Conjecture: a(n) is the number of compositions of n if all single instances of the part 1 are frozen ([1]). Example: The compositions enumerated by a(5) = 13 are 5; 4,[1]; 3,2; 2,3; 3,1,1; 1,3,1; 1,1,3; 2,2,[1]; 2,1,1,1; 1,2,1,1; 1,1,2,1; 1,1,1,2; 1,1,1,1,1. - Gregory L. Simay, Oct 27 2022
FORMULA
Equals row sums of triangle A221833.
EXAMPLE
G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 7*x^5 + 13*x^6 + 27*x^7 + 54*x^8 + ...
where
A(x) = x + x^2*(1-x^2)/(1-x) + x^3*(1-x^3)^2/(1-x)^2 + x^4*(1-x^4)^3/(1-x)^3 + ...
or, equivalently,
A(x) = x + x^2*(1+x) + x^3*(1+x+x^2)^2 + x^4*(1+x+x^2+x^3)^3 + ...
PROG
(PARI) {a(n)=polcoeff(sum(k=1, n, x^k*((1-x^k)/(1-x) +x*O(x^n))^(k-1)), n)}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 26 2013
STATUS
approved