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A194044 G.f. satisfies: A(x) = ( Sum_{n>=0} q^(n*(n+1)/2) )^4 where q=x*A(x)^2. 2
1, 4, 38, 472, 6685, 102340, 1649446, 27574712, 473750970, 8313682000, 148383186924, 2685209034144, 49154880453275, 908609423877476, 16935710715505290, 317951375873760120, 6006975695929624776, 114120962913881862036, 2178813296304338702764 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..18.

FORMULA

The g.f. A(x) satisfies:

(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)),

(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)),

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

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

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

due to q-series identities.

Self-convolution 2nd power equals A194042.

Self-convolution 4th root equals A194043.

EXAMPLE

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

where

(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 +... +...

(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) +...

(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) +...

(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 *...

(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 +...

Related expansions begin:

_ 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 +...

_ 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 +...

PROG

(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)}

(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)}

(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)}

(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)}

(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)}

(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)}

CROSSREFS

Cf. A194042, A194043.

Sequence in context: A220748 A192947 A234463 * A317605 A263376 A199813

Adjacent sequences:  A194041 A194042 A194043 * A194045 A194046 A194047

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 12 2011

STATUS

approved

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Last modified January 23 16:10 EST 2021. Contains 340385 sequences. (Running on oeis4.)