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%I #7 Feb 14 2019 22:09:49
%S 1,3,13,101,31,1785,10359,48837,361601,1518439,8107655,45047205,
%T 22174877,1409934297,7923887063,58640750927,268411539971,
%U 1425834725577,8083997355233,45849429916389,253004366571229,1487729015517467,8443414161401399,48141245001933381,155779268193228419,1569245091203776687,8970232353224094279,51314027859988631817,292380695300170801437,1682471873186160627609,10085943474769129981125,55294491352291112750853
%N Odd coefficients in Sum_{n>=0} (x^n + i)^n / (1 + i*x^n)^(n+1), in which the constant term is taken to be 1.
%C a(n) = A323689(n^2) for n >= 0.
%H Paul D. Hanna, <a href="/A323687/b323687.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = [x^(n^2)] Sum_{k>=0} (x^k + i)^k / (1 + i*x^k)^(k+1) for n > 0.
%F a(n) = [x^(n^2)] Sum_{k>=0} (x^k - i)^k / (1 - i*x^k)^(k+1) for n > 0.
%F a(n) = [x^(n^2)] Sum_{n>=0} (-i)^n * (1 + i*x^n)^(2*n+1) / (1 + x^(2*n))^(n+1) for n > 0.
%F a(n) = [x^(n^2)] Sum_{n>=0} i^n * (1 - i*x^n)^(2*n+1) / (1 + x^(2*n))^(n+1) for n > 0.
%e The generating function of A323689 is
%e G(x) = Sum_{n>=0} (x^n + i)^n / (1 + i*x^n)^(n+1),
%e in which the constant term is taken to be 1, so that
%e G(x) = 1 + 3*x - 14*x^3 + 13*x^4 + 22*x^5 - 30*x^7 - 82*x^8 + 101*x^9 - 46*x^11 + 170*x^12 + 54*x^13 - 524*x^15 + 31*x^16 + 70*x^17 - 78*x^19 + 442*x^20 + 1236*x^21 - 94*x^23 - 3204*x^24 + 1785*x^25 - 2428*x^27 + 842*x^28 + 118*x^29 - 126*x^31 + 6208*x^32 + 4228*x^33 - 14508*x^35 + 10359*x^36 + ...
%e This sequence gives the odd coefficients of x^n, which occur at n = k^2 for k >= 0.
%o (PARI) {A323689(n) = my(SUM = sum(m=0, n, (x^m + I +x*O(x^n))^m / (1 + I*x^m +x*O(x^n))^(m+1) ) ); polcoeff(1 + SUM - I^n/(1+I), n)}
%o {a(n) = A323689(n^2)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {A323689(n) = my(SUM = sum(m=0, n, (-I)^m*(1 + I*x^m +x*O(x^n))^(2*m+1) / (1 + x^(2*m) +x*O(x^n))^(m+1) ) ); polcoeff(1 + SUM - I^n/(1-I), n)}
%o {a(n) = A323689(n^2)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A323689.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 14 2019
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