|
|
EXAMPLE
|
G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 736*x^4 + 5632*x^5 + 49024*x^6 +...
where we have the following series identity:
A(x) = 1/((1+x)*(1-4*x)) + x/((1+x)^2*(1-8*x)) + x^2/((1+x)^3*(1-12*x))+ x^3/((1+x)^4*(1-16*x))+ x^4/((1+x)^5*(1-20*x)) + x^5/((1+x)^6*(1-24*x)) +...
is equal to
A(x) = 1 + 4*x*(1+x)/(1+4*x) + 2!*(4*x)^2*(1+x)^2/((1+4*x)*(1+8*x)) + 3!*(4*x)^3*(1+x)^3/((1+4*x)*(1+8*x)*(1+12*x)) + 4!*(4*x)^4*(1+x)^4/((1+4*x)*(1+8*x)*(1+12*x)*(1+16*x)) + 5!*(4*x)^5*(1+x)^5/((1+4*x)*(1+8*x)*(1+12*x)*(1+16*x)*(1+20*x)) +...
|
|
|
PROG
|
(PARI) {a(n)=polcoeff( sum(m=0, n, x^m/((1+x)^(m+1)*(1 - 4*(m+1)*x) +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff( sum(m=0, n, 4^m*m!*x^m*(1+x)^m/prod(k=1, m, 1+4*k*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, floor(n/2), sum(i=0, k, (-1)^i*binomial(k, i)*(k-i+1)^(n-k)*4^(n-k)))}
for(n=0, 30, print1(a(n), ", "))
|
