A324308 - OEIS

%I #11 Mar 12 2019 12:16:19

%S 1,2,18,560,68642,34215232,69224885118,564395560679680,

%T 18462516899827251202,2418515872778936155907072,

%U 1267759411784533615311138990078,2658525714048411732541898289615687680,22300920657959489563024348304023741280501762,748290577775586983580548698483165497859690809475072,100433695860518615261304925954158931177362414164610493153278,53919903879834365544910698161440014555837394338328471823987820789760

%N G.f.: Sum_{n>=0} (2^n + i)^n * x^n / (1 + 2^n*i*x)^(n+1), where i^2 = -1.

%C Notice that the imaginary components in the generating function vanish when expanded as a power series expansion in x.

%F G.f.: Sum_{n>=0} (2^n + i)^n * x^n / (1 + 2^n*i*x)^(n+1).

%F G.f.: Sum_{n>=0} (2^n - i)^n * x^n / (1 - 2^n*i*x)^(n+1).

%F a(n) = Sum_{k=0..n} i^k * binomial(n,k) * (2^n - 2^k*i)^(n-k).

%F a(n) = Sum_{k=0..n} (-i)^k * binomial(n,k) * (2^n + 2^k*i)^(n-k).

%e G.f.: A(x) = 1 + 2*x + 18*x^2 + 560*x^3 + 68642*x^4 + 34215232*x^5 + 69224885118*x^6 + 564395560679680*x^7 + 18462516899827251202*x^8 + ...

%e such that

%e A(x) = 1/(1+i*x) + (2 + i)*x/(1 + 2*i*x)^2 + (2^2 + i)^2*x^2/(1 + 2^2*i*x)^3 + (2^3 + i)^3*x^3/(1 + 2^3*i*x)^4 + (2^4 + i)^4*x^4/(1 + 2^4*i*x)^5 + (2^5 + i)^5*x^5/(1 + 2^5*i*x)^6 + (2^6 + i)^6*x^6/(1 + 2^6*i*x)^7 + ...

%e Also,

%e A(x) = 1/(1-i*x) + (2 - i)*x/(1 - 2*i*x)^2 + (2^2 - i)^2*x^2/(1 - 2^2*i*x)^3 + (2^3 - i)^3*x^3/(1 - 2^3*i*x)^4 + (2^4 - i)^4*x^4/(1 - 2^4*i*x)^5 + (2^5 - i)^5*x^5/(1 - 2^5*i*x)^6 + (2^6 - i)^6*x^6/(1 - 2^6*i*x)^7 + ...

%o (PARI) {a(n) = my(A = sum(m=0, n+1, (2^m + I)^m*x^m/(1 + I*2^m*x +x*O(x^n) )^(m+1) )); polcoeff(A, n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {a(n) = my(A = sum(m=0, n+1, (2^m - I)^m*x^m/(1 - I*2^m*x +x*O(x^n) )^(m+1) )); polcoeff(A, n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {a(n) = sum(k=0, n, I^k * binomial(n, k) * (2^n - 2^k*I)^(n-k) )}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {a(n) = sum(k=0, n, (-I)^k * binomial(n, k) * (2^n + 2^k*I)^(n-k) )}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A324306, A324307.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 09 2019