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%I #14 Dec 04 2022 23:03:45
%S 1,1,2,3,7,13,27,54,111,225,456,926,1877,3796,7671,15483,31212,62859,
%T 126484,254296,510892,1025765,2058395,4128578,8277344,16589180,
%U 33237163,66574351,133318484,266924608,534335692,1069492787,2140370294,4283071475,8570061106
%N G.f.: Sum_{n>=1} x^n * (1-x^n)^(n-1) / (1-x)^(n-1).
%C 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
%F Equals row sums of triangle A221833.
%e 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 + ...
%e where
%e 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 + ...
%e or, equivalently,
%e A(x) = x + x^2*(1+x) + x^3*(1+x+x^2)^2 + x^4*(1+x+x^2+x^3)^3 + ...
%o (PARI) {a(n)=polcoeff(sum(k=1,n,x^k*((1-x^k)/(1-x) +x*O(x^n))^(k-1)),n)}
%o for(n=1,40,print1(a(n),", "))
%Y Cf. A221833, A077229.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jan 26 2013
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