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%I #22 Feb 14 2019 21:05:22
%S 1,2,1,-8,0,18,11,-16,-44,0,36,-12,35,112,56,-144,-260,-88,48,-20,95,
%T 504,636,288,-578,-1016,-292,-216,-624,-210,73,-64,622,2892,4062,672,
%U -2916,-1508,1010,-80,-4894,-8618,-6809,-2728,46,6740,17502,19104,7040,-4820,-3898,1488,1080,-192,96,-8792,-37129,-57508,-40242,-12120,15902,62044,91082,41584,-51684,-50196,59462,122896,85616
%N G.f.: Sum_{n>=0} (x^(n+1) + i)^n / (1 + i*x^n)^(n+1), in which the constant term is taken to be 1.
%H Paul D. Hanna, <a href="/A323690/b323690.txt">Table of n, a(n) for n = 0..10100</a>
%F G.f.: i * Sum_{n>=0} (1 + i*x^(n+1))^n / (x^n + i)^(n+1), in which the constant term is taken to be 1.
%e G.f.: A(x) = 1 + 2*x + x^2 - 8*x^3 + 18*x^5 + 11*x^6 - 16*x^7 - 44*x^8 + 36*x^10 - 12*x^11 + 35*x^12 + 112*x^13 + 56*x^14 - 144*x^15 - 260*x^16 - 88*x^17 + 48*x^18 - 20*x^19 + 95*x^20 + 504*x^21 + 636*x^22 + 288*x^23 - 578*x^24 - 1016*x^25 - 292*x^26 - 216*x^27 - 624*x^28 - 210*x^29 + 73*x^30 + ...
%e which equals the following sum when expanded as a power series in x:
%e A(x) = 1/(1+i) + (x^2 + i)/(1 + i*x)^2 + (x^3 + i)^2/(1 + i*x^2)^3 + (x^4 + i)^3/(1 + i*x^3)^4 + (x^5 + i)^4/(1 + i*x^4)^5 + (x^6 + i)^5/(1 + i*x^5)^6 + (x^7 + i)^6/(1 + i*x^6)^7 + (x^8 + i)^7/(1 + i*x^7)^8 + (x^9 + i)^8/(1 + i*x^8)^9 + ...
%e where the coefficient of x^0 is taken to be 1.
%e The limit of the following sum expands into a power series in x with only real coefficients after the initial coefficient of x^0:
%e S(N) = Sum_{n=0..N} (x^(n+1) + i)^n / (1 + i*x^n)^(n+1) = i^N/(1+i) + 2*x + x^2 - 8*x^3 + 18*x^5 + 11*x^6 - 16*x^7 - 44*x^8 + 36*x^10 - 12*x^11 + 35*x^12 + ...
%e here, we ignore the coefficient of x^0 and set a(0) = 1.
%e TRIANGLE FORM.
%e This sequence may be written as a triangle that begins
%e 1, 2;
%e 1, -8, 0, 18;
%e 11, -16, -44, 0, 36, -12;
%e 35, 112, 56, -144, -260, -88, 48, -20;
%e 95, 504, 636, 288, -578, -1016, -292, -216, -624, -210;
%e 73, -64, 622, 2892, 4062, 672, -2916, -1508, 1010, -80, -4894, -8618;
%e -6809, -2728, 46, 6740, 17502, 19104, 7040, -4820, -3898, 1488, 1080, -192, 96, -8792;
%e -37129, -57508, -40242, -12120, 15902, 62044, 91082, 41584, -51684, -50196, 59462, 122896, 85616, 35300, 18822, 12072; ...
%e in which the first column consists of all the odd terms in this sequence.
%o (PARI) {a(n) = my(SUM = sum(m=0,n, (x^(m+1) + 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 for(n=0,100,print1(a(n),", "))
%Y Cf. A323691, A323557, A323689.
%K sign
%O 0,2
%A _Paul D. Hanna_, Feb 13 2019
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