OFFSET
0,2
COMMENTS
In general, we have the o.g.f. identity:
Sum_{n>=0} m^n * q^(n^2) * x^n/(1 - b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n ; here, q=2, m=2, b=1.
In general, we have the e.g.f. identity:
Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n! = Sum_{n>=0} (m*q^n + b)^n * x^n / n! ; here, q=2, m=2, b=1.
FORMULA
O.g.f.: Sum_{n>=0} 2^(n*(n+1)) * x^n/(1 - 2^n*x)^(n+1) = Sum_{n>=0} (2^(n+1) + 1)^n * x^n.
E.g.f.: Sum_{n>=0} 2^(n*(n+1)) * exp(2^n*x) * x^n / n! = Sum_{n>=0} (2^(n+1) + 1)^n * x^n / n!.
EXAMPLE
O.g.f.: A(x) = 1 + 5*x + 81*x^2 + 4913*x^3 + 1185921*x^4 + 1160290625*x^5 + 4608273662721*x^6 + 74051159531521793*x^7 + 4796659837465472798721*x^8 + ...
where
A(x) = 1/(1 - x) + 2^2*x/(1 - 2*x)^2 + 2^6*x^2/(1 - 2^2*x)^3 + 2^12*x^3/(1 - 2^3*x)^4 + 2^20*x^4/(1 - 2^4*x)^5 + 2^30*x^5/(1 - 2^5*x)^6 + ...
PROG
(PARI) {a(n, q, m, b) = (m*q^n + b)^n}
for(n=0, 15, print1(a(n, q=2, m=2, b=1), ", "))
(PARI) /* E.g.f. formula: */
{a(n, q, m, b) = polcoeff( sum(k=0, n, m^k * q^(k^2) * x^k / (1 - b*q^k*x +x*O(x^n))^(k+1)), n)}
for(n=0, 15, print1(a(n, q=2, m=2, b=1), ", "))
(PARI) /* E.g.f. formula: */
{a(n, q, m, b) = n! * polcoeff( sum(k=0, n, m^k * q^(k^2) * exp(b*q^k*x +x*O(x^n)) * x^k/k!), n)}
for(n=0, 15, print1(a(n, q=2, m=2, b=1), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 26 2020
STATUS
approved