A194044 - OEIS

%I #13 Mar 11 2015 01:35:24

%S 1,4,38,472,6685,102340,1649446,27574712,473750970,8313682000,

%T 148383186924,2685209034144,49154880453275,908609423877476,

%U 16935710715505290,317951375873760120,6006975695929624776,114120962913881862036,2178813296304338702764

%N G.f. satisfies: A(x) = ( Sum_{n>=0} q^(n*(n+1)/2) )^4 where q=x*A(x)^2.

%F The g.f. A(x) satisfies:

%F (1) A(x) = Sum_{n>=0} (2*n+1)*x^n*A(x)^(2*n)/(1 - x^(2*n+1)*A(x)^(4*n+2)),

%F (2) A(x)^2 = Sum_{n>=0} (n+1)^3*x^n*A(x)^(2*n)/(1 - x^(2*n+2)*A(x)^(4*n+4)),

%F (3) A(x) = Product_{n>=1} (1 + x^n*A(x)^(2*n))^4*(1 - x^(2*n)*A(x)^(4*n))^4,

%F (4) A(x) = exp( Sum_{n>=1} 4*(x^n*A(x)^(2*n)/(1 + x^n*A(x)^(2*n)))/n ),

%F (5) A(x/F(x)^8) = F(x)^4 where F(x) = Sum_{n>=0} x^(n*(n+1)/2),

%F due to q-series identities.

%F Self-convolution 2nd power equals A194042.

%F Self-convolution 4th root equals A194043.

%e G.f.: A(x) = 1 + 4*x + 38*x^2 + 472*x^3 + 6685*x^4 + 102340*x^5 +...

%e where

%e (0) A(x)^(1/4) = 1 + x*A(x)^2 + x^3*A(x)^6 + x^6*A(x)^12 + x^10*A(x)^20 + x^15*A(x)^30 + x^21*A(x)^42 +... +...

%e (1) A(x) = 1/(1-x*A(x)^2) + 3*x*A(x)^2/(1-x^3*A(x)^6) + 5*x^2*A(x)^4/(1-x^5*A(x)^10) + 7*x^3*A(x)^6/(1-x^7*A(x)^14) +...

%e (2) A(x)^2 = 1/(1-x^2*A(x)^4) + 8*x*A(x)^2/(1-x^4*A(x)^8) + 27*x^2*A(x)^4/(1-x^6*A(x)^12) + 64*x^3*A(x)^6/(1-x^8*A(x)^16) +...

%e (3) A(x) = (1+x*A(x)^2)^4*(1-x^2*A(x)^4)^4 * (1+x^2*A(x)^4)^4*(1-x^4*A(x)^8)^4 * (1+x^3*A(x)^6)^4*(1-x^6*A(x)^12)^4 * (1+x^4*A(x)^8)^4*(1-x^8*A(x)^16)^4 *...

%e (4) log(A(x)) = 4*x*A(x)^2/(1+x*A(x)^2) + 4*(x^2*A(x)^4/(1+x^2*A(x)^4))/2  + 4*(x^3*A(x)^6/(1+x^3*A(x)^6))/3 + 4*(x^4*A(x)^8/(1+x^4*A(x)^8))/4 +...

%e Related expansions begin:

%e _ A(x)^(1/4) = 1 + x + 8*x^2 + 93*x^3 + 1272*x^4 + 19058*x^5 + 302705*x^6 + 5007234*x^7 + 85341048*x^8 +...+ A194043(n)*x^n +...

%e _ A(x)^2 = 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 +...

%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)^(1/2); polcoeff(A, n)}

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(x*A^2+x*O(x^n))^(m*(m+1)/2))^4);polcoeff(A,n)}

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(2*m+1)*(x*A^2)^m/(1-(x*A^2+x*O(x^n))^(2*m+1))));polcoeff(A,n)}

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(m+1)^3*(x*A^2)^m/(1-(x*A^2+x*O(x^n))^(2*m+2)))^(1/2));polcoeff(A,n)}

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=prod(m=1,n,(1+(x*A^2)^m)*(1-(x*A^2)^(2*m)+x*O(x^n)))^4); polcoeff(A, n)}

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,4*(x*A^2)^m/(1+(x*A^2)^m+x*O(x^n))/m))); polcoeff(A, n)}

%Y Cf. A194042, A194043.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 12 2011