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A292177 G.f.: Limit_{K->oo} Sum_{n=-oo..+oo} x^(n-K) * (1 - x^n + n*(n+1)/6 * x^(n+K))^n.

%I A292177 #13 Sep 10 2017 17:36:11
%S A292177 1,0,4,0,5,0,20,-24,35,0,0,0,84,-160,200,0,-150,0,460,-560,286,0,140,
%T A292177 -200,455,-1440,2100,0,-2180,0,3840,-3080,969,-2240,2730,0,1330,-5824,
%U A292177 5320,0,-4235,0,16874,-21840,2300,0,18440,-784,-20175,-16320,37310,0,-945,-42240,49560,-25080,4495,0,7560,0,5456,-50400,102528,-120120,40810,0,135660,-52624,-221690,0,278256,0,9139,-364000,232750,-99792,211120,0,-106680,-100440,12341,0,537992,-628320,14190,-129920,563420,0,-195015,-480480,591100,-168640,18424,-1240320,2138640,0,-925120,-268224,-803250,0
%N A292177 G.f.: Limit_{K->oo} Sum_{n=-oo..+oo} x^(n-K) * (1 - x^n + n*(n+1)/6 * x^(n+K))^n.
%C A292177 Compare the g.f. to: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n  =  0.
%H A292177 Paul D. Hanna, <a href="/A292177/b292177.txt">Table of n, a(n) for n = 2..1002</a>
%F A292177 G.f.: Limit_{K->oo} Sum_{n=-oo..+oo} -(-1)^n * x^(n^2-n-K) / (1 - x^n + n*(n-1)/6 * x^K)^n.
%F A292177 a(p) = 0 for odd prime p (conjecture).
%e A292177 G.f.: A(x) = x^2 + 4*x^4 + 5*x^6 + 20*x^8 - 24*x^9 + 35*x^10 + 84*x^14 - 160*x^15 + 200*x^16 - 150*x^18 + 460*x^20 - 560*x^21 + 286*x^22 + 140*x^24 - 200*x^25 + 455*x^26 - 1440*x^27 + 2100*x^28 - 2180*x^30 + 3840*x^32 - 3080*x^33 + 969*x^34 - 2240*x^35 + 2730*x^36 + 1330*x^38 - 5824*x^39 + 5320*x^40 +...
%e A292177 such that the g.f. equals the limit of the sum, as K tends to infinity,
%e A292177 S(K) = Sum_{n=-oo..+oo} x^(n-K) * (1 - x^n + n*(n+1)/6 * x^(n+K))^n.
%e A292177 Illustration of the limit.
%e A292177 S(1) = x^2 + 1/3*x^3 + 112/27*x^4 + 113/81*x^5 + 467/81*x^6 - 938/729*x^7 +...
%e A292177 S(2) = x^2 + 13/3*x^4 + 175/27*x^6 + 1550/81*x^8 - 24*x^9 + 2777/81*x^10 +...
%e A292177 S(3) = x^2 + 4*x^4 + 1/3*x^5 + 5*x^6 + 4/3*x^7 + 544/27*x^8 - 77/3*x^9 +...
%e A292177 S(4) = x^2 + 4*x^4 + 16/3*x^6 + 64/3*x^8 - 24*x^9 + 904/27*x^10 +...
%e A292177 S(5) = x^2 + 4*x^4 + 5*x^6 + 1/3*x^7 + 20*x^8 - 68/3*x^9 + 35*x^10 +...
%e A292177 S(6) = x^2 + 4*x^4 + 5*x^6 + 61/3*x^8 - 24*x^9 + 109/3*x^10 - 5/3*x^12 +...
%e A292177 S(7) = x^2 + 4*x^4 + 5*x^6 + 20*x^8 - 71/3*x^9 + 35*x^10 + 4/3*x^11 +...
%e A292177 S(8) = x^2 + 4*x^4 + 5*x^6 + 20*x^8 - 24*x^9 + 106/3*x^10 + 4/3*x^12 +...
%e A292177 S(9) = x^2 + 4*x^4 + 5*x^6 + 20*x^8 - 24*x^9 + 35*x^10 + 1/3*x^11 +...
%e A292177 ...
%e A292177 At powers of 2, a(2^n) begins:
%e A292177 [1, 4, 20, 200, 3840, 102528, 8437440, 5275875200, 5635011683840, 2075681844543566848, 671078483184128826885120, ...].
%o A292177 (PARI) {a(n) = my(A=1,K=n); A = sum(m=-sqrtint(2*n+9), 2*n+1, x^(m-K) * (1 - x^m +m*(m+1)/6*x^(m+K) + O(x^(2*n+2)) )^m  ); polcoeff(A, n)}
%o A292177 for(n=2, 80, print1(a(n), ", "))
%Y A292177 Cf. A291937.
%K A292177 sign
%O A292177 2,3
%A A292177 _Paul D. Hanna_, Sep 10 2017