|
PROG
|
(PARI) /* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^(2*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* From closed formula: */
{a(n)=local(A=1); A= 1/sqrt((1 - x - x^3)^2 - 4*x^4 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* From a series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, (2*m)!/(m!)^2 * x^(3*m) / (1 - x + x^3 +x*O(x^n))^(2*m+1)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, x^m*(1-x^2)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(2*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\3, x^(3*m)*sum(k=0, n-3*m, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\3, x^(3*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* From exponential formula: */
{a(n)=local(A=1); A=exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k)*x^(2*k)) * x^m/m) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* From exponential formula: */
{a(n)=local(A=1); A=exp(sum(m=1, n, ((1+x)^(2*m) + (1-x)^(2*m))/2 * x^m/m) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* From formula for a(n): */
{a(n)=sum(k=0, n\2, binomial(n-2*k, k)^2)}
for(n=0, 40, print1(a(n), ", "))
|