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A192623
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G.f. satisfies: A(x) = Product_{n>=0} (1 + x^(n+1)*A(x)^(2*n))/(1 - x^(n+1)*A(x)^(2*n)).
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4
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1, 2, 4, 16, 70, 336, 1720, 9152, 50140, 280882, 1601496, 9263424, 54224312, 320611152, 1912003536, 11487287872, 69463274022, 422440713680, 2582081184572, 15853795192704, 97736576247976, 604744065493936, 3754311394271208, 23377930236777152
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OFFSET
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0,2
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COMMENTS
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Related q-series (Heine) identity:
1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n)); here q=x*A(x)^2, x=x, y=1, z=0.
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LINKS
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FORMULA
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G.f. satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^(n*(n+1)/2)*A(x)^(n*(n-1)) * Product_{k=0..n-1} (1 + x^k*A(x)^(2*k))/((1 - x^(k+1)*A(x)^(2*k))*(1 - x^(k+1)*A(x)^(2*k+2))), due to the Heine identity.
(2) A(x)^2 = 1 + Sum_{n>=1} x^n * Product_{k=0..n-1} (1 + x^k*A(x)^(2*k))^2/((1 - x^(k+1)*A(x)^(2*k))*(1 - x^(k+1)*A(x)^(2*k+2)), due to the Heine identity.
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 4*x^2 + 16*x^3 + 70*x^4 + 336*x^5 + 1720*x^6 +...
The g.f. A = A(x) satisfies the following relations:
(0) A = (1+x)/(1-x) * (1+x^2*A^2)/(1-x^2*A^2) * (1+x^3*A^4)/(1-x^3*A^4) * (1+x^4*A^6)/(1-x^4*A^6)*...
(1) A = 1 + 2*x/((1-x)*(1-x*A^2)) + 2*x^3*A^2*(1+x*A^2)/((1-x)*(1-x*A^2)*(1-x^2*A^2)*(1-x^2*A^4)) + 2*x^6*A^6*(1+x*A^2)*(1+x^2*A^4)/((1-x)*(1-x*A^2)*(1-x^2*A^2)*(1-x^2*A^4)*(1-x^3*A^4)*(1-x^3*A^6)) +...
(2) A^2 = 1 + 4*x/((1-x)*(1-x*A^2)) + 4*x^2*(1+x*A^2)^2/((1-x)*(1-x*A^2)*(1-x^2*A^2)*(1-x^2*A^4)) + 4*x^3*(1+x*A^2)^2*(1+x^2*A^4)^2/((1-x)*(1-x*A^2)*(1-x^2*A^2)*(1-x^2*A^4)*(1-x^3*A^4)*(1-x^3*A^6)) +...
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(k=0, n, (1+x^(k+1)*A^(2*k))/(1-x^(k+1)*(A+x*O(x^n))^(2*k)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^(m*(m+1)/2)*A^(m*(m-1))*prod(k=0, m-1, (1+x^k*A^(2*k))/((1-x^(k+1)*A^(2*k))*(1-x^(k+1)*A^(2*k+2) +x*O(x^n)))))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sqrt(1+sum(m=1, n, x^m*prod(k=0, m-1, (1+x^k*A^(2*k))^2/((1-x^(k+1)*A^(2*k) +x*O(x^n))*(1-x^(k+1)*A^(2*k+2))))))); polcoeff(A, n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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