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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 9*x^2 + 51*x^3 + 327*x^4 + 2252*x^5 + 16259*x^6 +...
where A = g.f. A(x) is given by the binomial series identity:
A(x) = 1/(1-x) + x*A(x)/(1-x)^3 * (1 + x)^2
+ x^2*A^2/(1-x)^5 * (1 + 2^2*x + x^2)^2
+ x^3*A^3/(1-x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3)^2
+ x^4*A^4/(1-x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2
+ x^5*A^5/(1-x)^11 * (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2 +...
equals the series
A(x) = 1/(1-x*A) + x/(1-x*A)^3 * (1 + x*A)^2
+ x^2/(1-x*A)^5 * (1 + 2^2*x*A + x^2*A^2)^2
+ x^3/(1-x*A)^7 * (1 + 3^2*x*A + 3^2*x^2*A^2 + x^3*A^3)^2
+ x^4/(1-x*A)^9 * (1 + 4^2*x*A + 6^2*x^2*A^2 + 4^2*x^3*A^3 + x^4*A^4)^2
+ x^5/(1-x*A)^11 * (1 + 5^2*x*A + 10^2*x^2*A^2 + 10^2*x^3*A^3 + 5^2*x^4*A^4 + x^5*A^5)^2 +...
We can also express the g.f. by another binomial series identity:
A(x) = 1 + x*(1 + A*(1+x)) + x^2*(1 + 2^2*A*(1+x) + A^2*(1+2^2*x+x^2))
+ x^3*(1 + 3^2*A*(1+x) + 3^2*A^2*(1+2^2*x+x^2) + A^3*(1+3^2*x+3^2*x^2+x^3))
+ x^4*(1 + 4^2*A*(1+x) + 6^2*A^2*(1+2^2*x+x^2) + 4^2*A^3*(1+3^2*x+3^2*x^2+x^3) + A^4*(1+4^2*x+6^2*x^2+4^2*x^3+x^4))
+ x^5*(1 + 5^2*A*(1+x) + 10^2*A^2*(1+2^2*x+x^2) + 10^2*A^3*(1+3^2*x+3^2*x^2+x^3) + 5^2*A^4*(1+4^2*x+6^2*x^2+4^2*x^3+x^4) + A^5*(1+5^2*x+10^2*x^2+10^2*x^3+5^2*x^4+x^5)) +...
equals the series
A(x) = 1 + x*(A + (1+x*A)) + x^2*(A^2 + 2^2*A*(1+x*A) + (1+2^2*x*A+x^2*A^2))
+ x^3*(A^3 + 3^2*A^2*(1+x*A) + 3^2*A*(1+2^2*x*A+x^2*A^2) + (1+3^2*x*A+3^2*x^2*A^2+x^3*A^3))
+ x^4*(A^4 + 4^2*A^3*(1+x*A) + 6^2*A^2*(1+2^2*x*A+x^2*A^2) + 4^2*A*(1+3^2*x*A+3^2*x^2*A^2+x^3*A^3) + (1+4^2*x*A+6^2*x^2*A^2+4^2*x^3*A^3+x^4*A^4))
+ x^5*(A^5 + 5^2*A^4*(1+x*A) + 10^2*A^3*(1+2^2*x*A+x^2*A^2) + 10^2*A^2*(1+3^2*x*A+3^2*x^2*A^2+x^3*A^3) + 5^2*A*(1+4^2*x*A+6^2*x^2*A^2+4^2*x^3*A^3+x^4*A^4) + (1+5^2*x*A+10^2*x^2*A^2+10^2*x^3*A^3+5^2*x^4*A^4+x^5*A^5)) +...
More explicitly, the g.f. can be written as the series:
A(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)
+ x*A(x)*(1 + 5*x + 13*x^2 + 25*x^3 + 41*x^4 + 61*x^5 +...)
+ x^2*A(x)^2*(1 + 13*x + 73*x^2 + 253*x^3 + 661*x^4 + 1441*x^5 +...)
+ x^3*A(x)^3*(1 + 25*x + 253*x^2 + 1445*x^3 + 5741*x^4 +...)
+ x^4*A(x)^4*(1 + 41*x + 661*x^2 + 5741*x^3 + 33001*x^4 +...)
+ x^5*A(x)^5*(1 + 61*x + 1441*x^2 + 17861*x^3 + 142001*x^4 +...) +...
where the coefficients form the square array A143007.
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PROG
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(PARI) /* By definition: */
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m * A^m / (1-x +x*O(x^n))^(2*m+1) * sum(k=0, m, binomial(m, k)^2*x^k)^2 )); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* By a binomial identity: */
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m/(1-x*A +x*O(x^n))^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * x^k * A^k)^2 )); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* By a binomial identity: */
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * A^k * sum(j=0, k, binomial(k, j)^2 * x^j)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* By a binomial identity: */
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * A^(m-k) * sum(j=0, k, binomial(k, j)^2 * x^j * A^j)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* From formula involving A143007: */
{A143007(n, k)=sum(j=0, n, binomial(n+j, 2*j)*binomial(2*j, j)^2*binomial(k+j, 2*j))}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, A143007(m-k, k) * A^k +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* From formula involving A143007: */
{A143007(n, k)=sum(j=0, n, binomial(n+j, 2*j)*binomial(2*j, j)^2*binomial(k+j, 2*j))}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*A^m*sum(k=0, n-m, A143007(m, k) * x^k +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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