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A196151 G.f. satisfies A(x) = Product_{n>=1} (1 + x^n*A(x)^2). 1
1, 1, 3, 11, 43, 179, 778, 3491, 16051, 75235, 358170, 1727124, 8418266, 41408344, 205289265, 1024737905, 5145933602, 25978844478, 131773584768, 671239285119, 3432304205872, 17611565623950, 90652384728648, 467963720803022, 2422110238147351 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..24.

FORMULA

G.f. satisfies:

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

(2) A(x) = Sum_{n>=0} x^(n*(3n+1)/2)*A(x)^(2*n)*(1 + x^(2n+1)*A(x)^2)*Product_{k=1..n} (1 + x^k*A(x)^2)/(1-x^k) due to Sylvester's identity.

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 43*x^4 + 179*x^5 + 778*x^6 +...

where

(0) A(x) = (1+x*A(x)^2) * (1+x^2*A(x)^2) * (1+x^3*A(x)^2) * (1+x^4*A(x)^2)*...

(1) A(x) = 1 + x*A(x)^2/(1-x) + x^3*A(x)^4/((1-x)*(1-x^2)) + x^6*A(x)^6/((1-x)*(1-x^2)*(1-x^3)) + x^10*A(x)^8/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) +...

(2) A(x) = (1+x*A(x)^2) + x^2*A(x)^2*(1 + x^3*A(x)^2)*(1+x*A(x)^2)/(1-x) + x^7*A(x)^4*(1 + x^5*A(x)^2)*(1+x*A(x)^2)*(1+x^2*A(x)^2)/((1-x)*(1-x^2)) + x^15*A(x)^6*(1 + x^7*A(x)^2)*(1+x*A(x)^2)*(1+x^2*A(x)^2)*(1+x^3*A(x)^2)/((1-x)*(1-x^2)*(1-x^3)) +...

PROG

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

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

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

CROSSREFS

Cf. A196150, A145267, A145268, A190822.

Sequence in context: A103821 A151095 A219262 * A149071 A149072 A284720

Adjacent sequences:  A196148 A196149 A196150 * A196152 A196153 A196154

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 28 2011

STATUS

approved

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Last modified August 20 19:51 EDT 2019. Contains 326155 sequences. (Running on oeis4.)