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 A196150 G.f. satisfies A(x) = 1/Product_{n>=1} (1 - x^n*A(x)^2). 5
 1, 1, 4, 18, 92, 505, 2922, 17541, 108270, 682823, 4380942, 28504466, 187636994, 1247375147, 8362420498, 56471709841, 383790966537, 2622982116829, 18016055333571, 124296340608870, 860986586024343, 5985590694574930, 41749023026002831 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS FORMULA G.f. satisfies: (1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(2*n) / Product_{k=1..n} (1-x^k) due to an identity of Euler. (2) A(x) = 1 + Sum_{n>=1} x^(n^2)*A(x)^(2*n)/[Product_{k=1..n} (1-x^k)*(1-x^k*A(x)^2)] due to Cauchy's identity. (3) A(x) = 1 + Sum_{n>=1} x^n*A(x)^2 / Product_{k=1..n} (1 - x^k*A(x)^2). EXAMPLE G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 92*x^4 + 505*x^5 + 2922*x^6 +... where (0) A(x) = 1/((1-x*A(x)^2) * (1-x^2*A(x)^2) * (1-x^3*A(x)^2) *...). (1) A(x) = 1 + x*A(x)^2/(1-x) + x^2*A(x)^4/((1-x)*(1-x^2)) + x^3*A(x)^6/((1-x)*(1-x^2)*(1-x^3)) +... (2) A(x) = 1 + x*A(x)^2/[(1-x)*(1-x*A(x)^2)] + x^4*A(x)^4/[(1-x)*(1-x^2)*(1-x*A(x)^2)*(1-x^2*A(x)^2)] + x^9*A(x)^6/[(1-x)*(1-x^2)*(1-x^3)*(1-x*A(x)^2)*(1-x^2*A(x)^2)*(1-x^3*A(x)^2)] +... (3) A(x) = 1 + x*A(x)^2/(1-x*A(x)^2) + x^2*A(x)^2/((1-x*A(x)^2)*(1-x^2*A(x)^2)) + x^3*A(x)^2/((1-x*A(x)^2)*(1-x^2*A(x)^2)*(1-x^3*A(x)^2)) +... PROG (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*A^2+x*O(x^n)))); polcoeff(A, n)} (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*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=1+sum(m=1, sqrtint(n+1), x^(m^2)*A^(2*m)/prod(k=1, m, (1-x^k)*(1-x^k*A^2+x*O(x^n))))); polcoeff(A, n)} (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^2/prod(k=1, m, (1-x^k*A^2+x*O(x^n))))); polcoeff(A, n)} CROSSREFS Cf. A196151, A145268, A145267, A206639, A206637. Sequence in context: A269450 A206639 A172964 * A185298 A255397 A081923 Adjacent sequences:  A196147 A196148 A196149 * A196151 A196152 A196153 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 28 2011 STATUS approved

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