OFFSET
0,2
COMMENTS
Note that the generating function, which involves an irrational constant, expands as a power series in x that consists solely of integer coefficients.
FORMULA
G.f.: Sum_{n>=0} (2^n + q)^n * x^n / (1 + 2^n*q*x)^(n+1), where q = sqrt(2).
G.f.: Sum_{n>=0} (2^n - q)^n * x^n / (1 - 2^n*q*x)^(n+1), where q = sqrt(2).
a(n) = Sum_{k=0..n} binomial(n,k) * (2^n - 2^k*q)^(n-k) * q^k, where q = sqrt(2).
a(n) = Sum_{k=0..n} binomial(n,k) * (2^n + 2^k*q)^(n-k) * (-q)^k, where q = sqrt(2).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 12*x^2 + 416*x^3 + 59528*x^4 + 32265472*x^5 + 67721176080*x^6 + 560073696753664*x^7 + 18415259721236185120*x^8 + ...
Let q = sqrt(2), then
A(x) = 1/(1+q*x) + (2 + q)*x/(1 + 2*q*x)^2 + (2^2 + q)^2*x^2/(1 + 2^2*q*x)^3 + (2^3 + q)^3*x^3/(1 + 2^3*q*x)^3 + (2^4 + q)^4*x^4/(1 + 2^4*q*x)^5 + (2^5 + q)^5*x^5/(1 + 2^5*q*x)^6 + (2^6 + q)^6*x^6/(1 + 2^6*q*x)^7 + ...
and also
A(x) = 1/(1-q*x) + (2 - q)*x/(1 - 2*q*x)^2 + (2^2 - q)^2*x^2/(1 - 2^2*q*x)^3 + (2^3 - q)^3*x^3/(1 - 2^3*q*x)^3 + (2^4 - q)^4*x^4/(1 - 2^4*q*x)^5 + (2^5 - q)^5*x^5/(1 - 2^5*q*x)^6 + (2^6 - q)^6*x^6/(1 - 2^6*q*x)^7 + ...
PROG
(PARI) {a(n) = my(q=sqrt(2), A = sum(m=0, n+1, (2^m + q)^m*x^m/(1 + 2^m*q*x +x*O(x^n) )^(m+1) )); round( polcoeff(A, n) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(q=sqrt(2), A = sum(m=0, n+1, (2^m - q)^m*x^m/(1 - 2^m*q*x +x*O(x^n) )^(m+1) )); round( polcoeff(A, n) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(q=sqrt(2)); round( sum(k=0, n, binomial(n, k) * (2^n - 2^k*q)^(n-k) * q^k ) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(q=sqrt(2)); round( sum(k=0, n, binomial(n, k) * (2^n + 2^k*q)^(n-k) * (-q)^k ) )}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 13 2019
STATUS
approved