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%I #18 Mar 12 2015 20:16:11
%S 1,1,8,93,1272,19058,302705,5007234,85341048,1488247757,26428516221,
%T 476280318599,8688447566960,160129594727162,2977120816382182,
%U 55769513231690143,1051605638580341912,19944384212454080598,380202253988763211734,7281079151205266545359
%N G.f. satisfies: A(x) = Sum_{n>=0} q^(n*(n+1)/2) where q = x*A(x)^8.
%F The g.f. A(x) satisfies:
%F (1) A(x)^2 = Sum_{n>=0} x^(n*(n+1)/2) * A(x)^((2*n+1)^2),
%F (2) A(x)^4 = Sum_{n>=0} (2*n+1)*x^n*A(x)^(8*n)/(1 - x^(2*n+1)*A(x)^(16*n+8)),
%F (3) A(x)^8 = Sum_{n>=0} (n+1)^3*x^n*A(x)^(8*n)/(1 - x^(2*n+2)*A(x)^(16*n+16)),
%F (4) A(x) = Product_{n>=1} (1 + x^n*A(x)^(8*n))*(1 - x^(2*n)*A(x)^(16*n)),
%F (5) A(x) = exp( Sum_{n>=1} (x^n*A(x)^(8*n)/(1 + x^n*A(x)^(8*n)))/n ),
%F (6) A(x/F(x)^8) = F(x) where F(x) = Sum_{n>=0} x^(n*(n+1)/2),
%F due to q-series identities.
%F Self-convolution 8th power equals A194042.
%F Self-convolution 4th power equals A194044.
%e G.f.: A(x) = 1 + x + 8*x^2 + 93*x^3 + 1272*x^4 + 19058*x^5 + 302705*x^6 +...
%e Let q = x*A(x)^8, then g.f. A(x) satisfies:
%e (0) A(x) = 1 + q + q^3 + q^6 + q^10 + q^15 + q^21 + q^28 + q^36 +...
%e The g.f. A(x) also satisfies:
%e (1) A(x)^2 = A(x) + x*A(x)^9 + x^3*A(x)^25 + x^6*A(x)^49 + x^10*A(x)^81 + x^15*A(x)^121 + x^21*A(x)^169 + x^28*A(x)^225 +...
%e (2) A(x)^4 = 1/(1-x*A(x)^8) + 3*x*A(x)^8/(1-x^3*A(x)^24) + 5*x^2*A(x)^16/(1-x^5*A(x)^40) + 7*x^3*A(x)^24/(1-x^7*A(x)^56) +...
%e (3) A(x)^8 = 1/(1-x^2*A(x)^16) + 8*x*A(x)^8/(1-x^4*A(x)^32) + 27*x^2*A(x)^16/(1-x^6*A(x)^48) + 64*x^3*A(x)^24/(1-x^8*A(x)^64) +...
%e (4) A(x) = (1+x*A(x)^8)*(1-x^2*A(x)^16) * (1+x^2*A(x)^16)*(1-x^4*A(x)^32) * (1+x^3*A(x)^24)*(1-x^6*A(x)^48) * (1+x^4*A(x)^32)*(1-x^8*A(x)^64) *...
%e (5) log(A(x)) = x*A(x)^8/(1+x*A(x)^8) + (x^2*A(x)^16/(1+x^2*A(x)^16))/2 + (x^3*A(x)^24/(1+x^3*A(x)^24))/3 + (x^4*A(x)^32/(1+x^4*A(x)^32))/4 +...
%e Related expansions begin:
%e _ A(x)^2 = 1 + 2*x + 17*x^2 + 202*x^3 + 2794*x^4 + 42148*x^5 + 672527*x^6 + 11161398*x^7 + 190702616*x^8 +...
%e _ A(x)^4 = 1 + 4*x + 38*x^2 + 472*x^3 + 6685*x^4 + 102340*x^5 + 1649446*x^6 + 27574712*x^7 + 473750970*x^8 +...+ A194044(n)*x^n +...
%e _ A(x)^8 = 1 + 8*x + 92*x^2 + 1248*x^3 + 18590*x^4 + 294032*x^5 + 4848456*x^6 + 82433472*x^7 + 1434755717*x^8 +...+ A194042(n)*x^n +...
%e _ log(A(x)) = x + 15*x^2/2 + 256*x^3/3 + 4619*x^4/4 + 85956*x^5/5 + 1631376*x^6/6 + 31387840*x^7/7 + 609993603*x^8/8 +...
%o (PARI) {a(n)=local(A=1+x,T=sum(m=0,sqrtint(2*n+1),x^(m*(m+1)/2))+x*O(x^n)); A=(serreverse(x/T^8)/x)^(1/8); polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,sqrt(2*n+1),x^(m*(m+1)/2)*(A+x*O(x^n))^((2*m+1)^2-1)));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(2*m+1)*x^m*A^(8*m)/(1-x^(2*m+1)*(A+x*O(x^n))^(16*m+8)))^(1/4));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(m+1)^3*x^m*A^(8*m)/(1-x^(2*m+2)*(A+x*O(x^n))^(16*m+16)))^(1/8));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x); for(i=1,n,A=prod(m=1,n,(1+(x*A^8)^m)*(1-(x*A^8)^(2*m)+x*O(x^n)))); polcoeff(A, n)}
%o (PARI) {a(n)=local(A=1+x); for(i=1,n,A=exp(sum(m=1,n,(x*A^8)^m/(1+(x*A^8)^m+x*O(x^n))/m))); polcoeff(A, n)}
%Y Cf. A194042, A194044.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 12 2011
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