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A193115 G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(2*n+1). 5
1, 1, 3, 12, 54, 265, 1373, 7388, 40888, 231250, 1330618, 7764670, 45841323, 273316120, 1643345418, 9953021248, 60665811025, 371850104167, 2290623433302, 14173331572490, 88049709138896, 548978010516319, 3434070688405887, 21545961024510032 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..23.

FORMULA

The g.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} (-x)^n*A(x)^(2*n+1) * Product_{k=1..n} (1 + x^(4*k-3)*A(x)^2)/(1 + x^(4*k-1)*A(x)^2);

(2) 1 = A(x)/(1+ x*A(x)^2/(1- x*(1-x^2)*A(x)^2/(1+ x^5*A(x)^2/(1- x^3*(1-x^4)*A(x)^2/(1+ x^9*A(x)^2/(1- x^5*(1-x^6)*A(x)^2/(1+ x^13*A(x)^2/(1- x^7*(1-x^8)*A(x)^2/(1- ...))))))))) (continued fraction);

due to identities of a partial elliptic theta function.

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 54*x^4 + 265*x^5 + 1373*x^6 +...

which satisfies:

1 = A(x) - x*A(x)^3 + x^4*A(x)^5 - x^9*A(x)^7 + x^16*A(x)^9 -+...

Related expansions.

A(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 270*x^4 + 1398*x^5 + 7518*x^6 +...

A(x)^5 = 1 + 5*x + 25*x^2 + 130*x^3 + 695*x^4 + 3816*x^5 +...

PROG

(PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(#A)+1, (-x)^(m^2)*Ser(A)^(2*m+1) ), #A-1)); if(n<0, 0, A[n+1])}

CROSSREFS

Cf. A193111, A193112, A193113, A193114, A193116.

Sequence in context: A200740 A177133 A186241 * A270489 A335819 A263853

Adjacent sequences:  A193112 A193113 A193114 * A193116 A193117 A193118

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 16 2011

STATUS

approved

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Last modified August 1 18:47 EDT 2021. Contains 346402 sequences. (Running on oeis4.)