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%I #32 Nov 10 2020 03:22:38
%S 1,1,3,9,35,157,799,4525,28131,189961,1381731,10752725,89024543,
%T 780441893,7215286619,70101824033,713576518099,7589578829637,
%U 84143768397175,970341357523749,11616886603985203,144132396583436329,1850345648687946571,24543468645365759797
%N G.f.: Sum_{n>=0} ( Product_{k=1..n} 1 + x*(1+x)^k ) / 2^(n+1).
%H Paul D. Hanna, <a href="/A338793/b338793.txt">Table of n, a(n) for n = 0..300</a>
%F G.f.: Sum_{n>=0} ( Product_{k=1..n} 1 + x*(1+x)^k ) / 2^(n+1).
%F G.f.: Sum_{n>=0} x^n * (1+x)^(n*(n+1)/2) / ( Product_{k=0..n} 2 - (1+x)^k ).
%e G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 35*x^4 + 157*x^5 + 799*x^6 + 4525*x^7 + 28131*x^8 + 189961*x^9 + 1381731*x^10 + 10752725*x^11 + 89024543*x^12 + ...
%e where
%e A(x) = 1/2 + (1 + x*(1+x))/2^2 + (1 + x*(1+x))*(1 + x*(1+x)^2)/2^3 + (1 + x*(1+x))*(1 + x*(1+x)^2)*(1 + x*(1+x)^3)/2^4 + (1 + x*(1+x))*(1 + x*(1+x)^2)*(1 + x*(1+x)^3)*(1 + x*(1+x)^4)/2^5 + ... + (Product_{k=1..n} 1 + x*(1+x)^k)/2^(n+1) + ...
%e Also,
%e A(x) = 1 + x*(1+x)/(2 - (1+x)) + x^2*(1+x)^3/((2 - (1+x))*(2 - (1+x)^2)) + x^3*(1+x)^6/((2 - (1+x))*(2 - (1+x)^2)*(2 - (1+x)^3)) + x^4*(1+x)^10/((2 - (1+x))*(2 - (1+x)^2)*(2 - (1+x)^3)*(2 - (1+x)^4)) + x^5*(1+x)^15/((2 - (1+x))*(2 - (1+x)^2)*(2 - (1+x)^3)*(2 - (1+x)^4)*(2 - (1+x)^5)) + ... + x^n*(1+x)^(n*(n+1)/2)/(Product_{k=0..n} 2 - (1+x)^k) + ...
%o (PARI) {a(n) = polcoeff( sum(m=0,n, x^m * (1+x +x*O(x^n))^(m*(m+1)/2) / prod(k=0,m, 2 - (1+x +x*O(x^n) )^k ) ), n)}
%o for(n=0,30,print1(a(n),", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 09 2020
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