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%I #13 Feb 15 2021 22:41:27
%S 1,3,12,22,63,57,181,174,318,302,714,444,1177,852,1239,1349,2598,1440,
%T 3586,2226,3381,3282,6246,3174,6980,5343,7434,6031,12111,5076,14638,
%U 9636,12381,11513,16125,9441,24115,15765,19743,14982,32076,13317,36726,21783,25062
%N G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: [Sum_{n>=0} x^n/(1 - x^(n+1))]^3 = Sum_{n>=0} a(n)*x^n/(1 - x^(n+1))^3.
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F (1) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^3 = Sum_{n>=0} a(n) * x^n / (1 - x^(n+1))^3.
%F (2) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^3 = Sum_{n>=0} (n+1)*(n+2)/2 * x^n * A( x^(n+1) ).
%e A(x) = 1 + 3*x + 12*x^2 + 22*x^3 + 63*x^4 + 57*x^5 + 181*x^6 + 174*x^7 + 318*x^8 + 302*x^9 + 714*x^10 + 444*x^11 + 1177*x^12 + ...
%e such that
%e D(x)^3 = 1/(1-x)^3 + 3*x/(1-x^2)^3 + 12*x^2/(1-x^3)^3 + 22*x^3/(1-x^4)^3 + 63*x^4/(1-x^5)^3 + 57*x^5/(1-x^6)^3 + ... + a(n)*x^n/(1-x^(n+1))^3 + ...
%e and
%e D(x)^3 = A(x) + 3*x*A(x^2) + 6*x^2*A(x^3) + 10*x^3*A(x^4) + 15*x^4*A(x^5) + 21*x^5*A(x^6) + 28*x^6*A(x^7) + ... + (n+1)*(n+2)/2*x^n*A(x^(n+1)) + ...
%e where
%e D(x)^3 = 1 + 6*x + 18*x^2 + 41*x^3 + 78*x^4 + 132*x^5 + 209*x^6 + 306*x^7 + 435*x^8 + 591*x^9 + 780*x^10 + 1008*x^11 + ... + A191829(n+1)*x^n + ...
%e D(x) = 1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 4*x^7 + 3*x^8 + 4*x^9 + 2*x^10 + 6*x^11 + 2*x^12 + ... + A000005(n+1)*x^n + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( sum(n=0,#A, x^n/(1 - x^(n+1) +x*O(x^#A)) )^3 - sum(n=0,#A-1,A[n+1]*x^n/(1 - x^(n+1) + x*O(x^#A))^3 ), #A-1) );A[n+1]}
%o for(n=0,100,print1(a(n),", "))
%Y Cf. A341373, A341375, A191829, A000005.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 11 2021