A221834 G.f.: Sum_{n>=1} x^n * (1-x^n)^(n-1) / (1-x)^(n-1).

1, 1, 2, 3, 7, 13, 27, 54, 111, 225, 456, 926, 1877, 3796, 7671, 15483, 31212, 62859, 126484, 254296, 510892, 1025765, 2058395, 4128578, 8277344, 16589180, 33237163, 66574351, 133318484, 266924608, 534335692, 1069492787, 2140370294, 4283071475, 8570061106

Offset: 1

Author

Paul D. Hanna, Jan 26 2013

Keywords

• nonn

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

Examples

			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 + ...
		

Crossrefs

Cf. A221833, A077229.

Programs

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),", "))

Formula

Equals row sums of triangle A221833.