A323685 - OEIS

%I #7 Feb 12 2019 16:08:42

%S 1,1,2,15,46,207,1201,6283,36746,235463,1553311,10803272,79101355,

%T 602629168,4775430042,39306129479,334963829368,2949993280119,

%U 26808687950425,250987986961396,2417350292179932,23922186855590303,242961589181754713,2529832992050854458,26980268905373556691,294452742973863998098,3285813227185410286520

%N G.f.: Sum_{n>=0} x^n * ((1+x)^n + sqrt(5)*i)^n / (1 + sqrt(5)*i*x*(1+x)^n)^(n+1), where i^2 = -1.

%C Note that the generating function expands into a power series in x with only real integer coefficients.

%H Paul D. Hanna, <a href="/A323685/b323685.txt">Table of n, a(n) for n = 0..300</a>

%F G.f.: Sum_{n>=0} x^n * ((1+x)^n + sqrt(5)*i)^n / (1 + sqrt(5)*i*x*(1+x)^n)^(n+1).

%F G.f.: Sum_{n>=0} x^n * ((1+x)^n - sqrt(5)*i)^n / (1 - sqrt(5)*i*x*(1+x)^n)^(n+1).

%e G.f.: A(x) = 1 + x + 2*x^2 + 15*x^3 + 46*x^4 + 207*x^5 + 1201*x^6 + 6283*x^7 + 36746*x^8 + 235463*x^9 + 1553311*x^10 + 10803272*x^11 + ...

%e Let r = sqrt(5)*i, so that r^2 = -5, then

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

%e also,

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

%o (PARI) {a(n) = my(r = sqrt(5)*I, A = sum(m=0,n+2, x^m*((1+x +x*O(x^n))^m + r)^m/(1 + r*x*(1+x +x*O(x^n))^m)^(m+1) )); round(polcoeff(A,n))}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n) = my(r = sqrt(5)*I, A = sum(m=0,n+2, x^m*((1+x +x*O(x^n))^m - r)^m/(1 - r*x*(1+x +x*O(x^n))^m)^(m+1) )); round(polcoeff(A,n))}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A323681, A323682, A323683, A323684.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 12 2019