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%I #15 Nov 22 2014 18:16:13
%S 1,1,-4,12,-120,1080,-14400,241920,-4233600,103783680,-2408071680,
%T 74468782080,-2230518850560,81262621440000,-2999630643609600,
%U 125381910682828800,-5464949027586048000,258986473607083622400,-12924791918438400000000,688106822684149835366400
%N E.g.f. A(x) satisfies: (A(x)^2 - 4*x)^5 = (2 - A(x)^5)^2.
%H Vaclav Kotesovec, <a href="/A249788/b249788.txt">Table of n, a(n) for n = 0..360</a>
%F E.g.f.: (1 + 5*Series_Reversion(G(x)))^(1/5), where G(x) = ((1+5*x)^(2/5) - (1-5*x)^(2/5))/4 = x + Sum_{n>=1} x^(2*n+1)/(2*n+1)! * Product_{k=0..n-1} (10*k+3)*(10*k+8).
%F E.g.f. A(x) satisfies:
%F (1) A(x)^5 + A(-x)^5 = 2.
%F (2) A(x)^2 - A(-x)^2 = 4*x.
%F (3) x = (A(x)^2 - (2 - A(x)^5)^(2/5))/4.
%F (4) A(x) = B(x^2) + x/B(x^2), where B(x^2) = (A(x) + A(-x))/2.
%F a(n) ~ (-1)^(n+1) * 2^(8*n/5-3/10) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Nov 15 2014
%e E.g.f.: A(x) = 1 + x - 4*x^2/2! + 12*x^3/3! - 120*x^4/4! + 1080*x^5/5! - 14400*x^6/6! + 241920*x^7/7! - 4233600*x^8/8! + 103783680*x^9/9! +...
%e where
%e A(x)^2 = 1 + 2*x - 6*x^2/2! - 48*x^4/4! + 1440*x^6/6! + 1088640*x^8/8! + 500048640*x^10/10! + 254062448640*x^12/12! +...
%e A(x)^5 = 1 + 5*x - 120*x^3/3! + 720*x^5/5! + 302400*x^7/7! + 96163200*x^9/9! + 37362124800*x^11/11! + 17236393574400*x^13/13! +...
%e Thus the coefficients of odd powers of x in A(x)^2 equal zero:
%e [1, 2, -6, 0, -48, 0, 1440, 0, 1088640, 0, 500048640, 0, ...],
%e while the coefficients of even powers of x in A(x)^5 equal zero:
%e [1, 5, 0, -120, 0, 720, 0, 302400, 0, 96163200, 0, 37362124800, 0, ...],
%e after a few initial terms.
%e EXPLICIT FORMULA.
%e Let G(x) = ((1+5*x)^(2/5) - (1-5*x)^(2/5))/4, which begins
%e G(x) = x + 24*x^3/3! + 5616*x^5/5! + 3616704*x^7/7! + 4535346816*x^9/9! + 9360955828224*x^11/11! +...+ [Product_{k=0..n-1} (10*k+3)*(10*k+8)]*x^(2*n+1)/(2*n+1)! +...
%e then (A(x)^5 - 1)/5 = Series_Reversion(G(x)).
%o (PARI) /* Explicit formula: */
%o {a(n)=local(A, X=x+x^2*O(x^n), G=((1+5*X)^(2/5) - (1-5*X)^(2/5))/4);
%o A=(1 + 5*serreverse(G))^(1/5); n!*polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* Formula using series expansion: */
%o {a(n)=local(A, G=x + sum(m=1, n\2+1, x^(2*m+1)/(2*m+1)!*prod(k=0, m-1, (10*k+3)*(10*k+8)) +x^2*O(x^n)));
%o A=(1 + 5*serreverse(G))^(1/5); n!*polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* Alternating zero coefficients in A(x)^2 and A(x)^5: */
%o {a(n)=local(A=[1, 1], E=1, M); for(i=1, n, A=concat(A, 0); M=#A;
%o E=sum(m=0, M-1, A[m+1]*x^m/m!)+x*O(x^M);
%o A[M]=if(M%2==0, -(M-1)!*Vec(E^2/2)[M], -(M-1)!*Vec(E^5/5)[M])); A[n+1]}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A249787 (dual), A249785, A249786, A249789.
%K sign
%O 0,3
%A _Paul D. Hanna_, Nov 15 2014
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