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%I #9 Oct 13 2023 11:11:48
%S 2,4,6,6,10,78,412,1394,3312,6416,17454,83334,384284,1377888,3931286,
%T 10234748,31776266,127848076,527518582,1910397078,6035143914,
%U 18202417974,60151348904,226355566282,874920531958,3166323335574,10599244540550,34588365630694,118339356017608
%N a(n) = coefficient of x^n in A(x) where 1 = Sum_{n=-oo..+oo} (x^n - x*A(x))^n.
%C All terms are even: a(n) = 2 * A355868(n) for n >= 0.
%C Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
%H Paul D. Hanna, <a href="/A359671/b359671.txt">Table of n, a(n) for n = 0..400</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F (1) 1 = Sum_{n=-oo..+oo} (x^n - x*A(x))^n.
%F (2) 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (x^n + A(x))^n.
%F (3) 0 = Sum_{n=-oo..+oo} (-1)^n * (x^n - x*A(x))^(n-1).
%F (4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (x^n + x*A(x))^(n+1).
%F (5) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 - A(x)*x^(n+1))^n.
%F (6) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 + A(x)*x^(n+1))^(n+1).
%F (7) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)*x^n)^n.
%e G.f.: A(x) = 2 + 4*x + 6*x^2 + 6*x^3 + 10*x^4 + 78*x^5 + 412*x^6 + 1394*x^7 + 3312*x^8 + 6416*x^9 + 17454*x^10 + 83334*x^11 + 384284*x^12 + ...
%e SPECIFIC VALUES.
%e A(x) = 3 at x = 0.1794935271005324391410493657541129782265990045275870...
%e A(x) = 4 at x = 0.2492900841034309263190875287839455698977108450414094...
%e A(x) = 5 at x = 0.2676600392887397049709560009239544652896107097280049...
%o (PARI) {a(n) = my(A=[2]); for(i=1, n, A=concat(A, 0);
%o A[#A] = polcoeff( sum(m=-#A, #A, (x^m - x*Ser(A))^m ), #A)); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=[2]); for(i=1, n, A=concat(A, 0);
%o A[#A] = polcoeff( sum(m=-#A, #A, x^(2*m+1) * (x^m + Ser(A))^m ), #A)); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=[2]); for(i=1, n, A=concat(A, 0);
%o A[#A] = polcoeff( sum(m=-#A, #A, x^(m^2)/(1 - Ser(A)*x^(m+1))^m ), #A)); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=[2]); for(i=1, n, A=concat(A, 0);
%o A[#A] = polcoeff( sum(m=-#A, #A, x^(m^2)/(1 + Ser(A)*x^(m+1))^(m+1) ), #A)); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A355868.
%K nonn
%O 0,1
%A _Paul D. Hanna_, Jan 10 2023