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%I #13 Feb 21 2019 21:02:29
%S 1,0,2,0,2,0,2,0,33284,0,2,0,883460,0,2,0,2,0,979339580806,0,2,0,
%T 115667460,0,2,0,2,0,799941380,0,23305156883711135492,0,
%U 38029776687864458395142,0,4012026683676222852,0,2,0,44190267844175072072966,0,2,0,20586734084,0,1978148160774064661252,0,29798631414807274867656075256452,0,59960389124,0,1826037115915435009712832114231669892,0,113795930884,0,2,0,2,0,272716398084,0,3420684050214265377966852
%N a(n) = [x^(n^2)] Sum_{m>=0} x^m * (x^(m+1) + i)^m / (1 + i*x^(m+1))^(m+1), for n >= 0.
%C a(n) = A324300(n^2) for n >= 0.
%H Paul D. Hanna, <a href="/A324301/b324301.txt">Table of n, a(n) for n = 0..180</a>
%F a(n) = [x^(n^2)] Sum_{n>=0} x^n * (x^(n+1) + i)^n / (1 + i*x^(n+1))^(n+1).
%F a(n) = [x^(n^2)] Sum_{n>=0} x^n * (x^(n+1) - i)^n / (1 - i*x^(n+1))^(n+1).
%F a(n) = [x^(n^2)] Sum_{n>=0} (i*x)^n * (1 - i*x^(n+1))^(2*n+1) / (1 + x^(2*n+2))^(n+1).
%F a(n) = [x^(n^2)] Sum_{n>=0} (-i*x)^n * (1 + i*x^(n+1))^(2*n+1) / (1 + x^(2*n+2))^(n+1).
%F a(2*n+1) = 0 for n >= 0.
%e The g.f. of A324300 is given by
%e G(x) = Sum_{n>=0} x^n * (x^(n+1) + i)^n / (1 + i*x^(n+1))^(n+1)
%e where
%e G(x) = 1 - 2*x^2 + 3*x^3 + 2*x^4 - 2*x^6 - 14*x^7 + 15*x^8 - 2*x^10 + 22*x^11 + 2*x^12 - 84*x^14 + 33*x^15 + 2*x^16 - 2*x^18 + 38*x^19 + 172*x^20 - 2*x^22 - 508*x^23 + 323*x^24 - 292*x^26 + 54*x^27 + 2*x^28 - 2*x^30 + 1088*x^31 + 444*x^32 - 2580*x^34 + 1753*x^35 + ...
%e such that
%e G(x) = 1/(1+i*x) + x*(x^2+i)/(1+i*x^2)^2 + x^2*(x^3+i)^2/(1+i*x^3)^3 + x^3*(x^4+i)^3/(1+i*x^4)^4 + x^4*(x^5+i)^4/(1+i*x^5)^5 + x^5*(x^6+i)^5/(1+i*x^6)^6 + x^6*(x^7+i)^6/(1+i*x^7)^7 + x^7*(x^8+i)^7/(1+i*x^8)^8 + ...
%e also
%e G(x) = (1-i*x)/(1+x^2) + i*x*(1-i*x^2)^3/(1+x^4)^2 - x^2*(1-i*x^3)^5/(1+x^6)^3 - i*x^3*(1-i*x^4)^7/(1+x^8)^4 + x^4*(1-i*x^5)^9/(1+x^10)^5 + i*x^5*(1-i*x^6)^11/(1+x^12)^6 + i*x^6*(1-i*x^7)^11/(1+x^14)^7 + ...
%e Note that the imaginary components in the above sums vanish.
%o (PARI) {A324300(n) = my(SUM = sum(m=0, n, x^m*(x^(m+1) + I +x*O(x^n))^m / (1 + I*x^(m+1) +x*O(x^n))^(m+1) ) ); polcoeff(SUM, n)}
%o {a(n) = A324300(n^2)}
%o for(n=0,60, print1(a(n),", "))
%Y Cf. A324300, A324302, A324303.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 21 2019
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