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%I #5 Oct 13 2016 00:07:23
%S 1,2,35,812,21359,623244,18568947,638475040,13249877870,2024051330358,
%T -355660668390645,130426094235366208,-54120354853298252400,
%U 27045033537893084984896,-15918675371944450999486319,10905983125914263654567255488,-8603776324190250513027830925715,7743542274281960968631431340349870,-7886327135586560316787947739703112447,9023297352140462809043434127286176617288,-11524615288427474577090651960651636283169590
%N G.f. A(x) satisfies: Series_Reversion( A(x) + A(x)^4 ) = A(x) - A(x)^4.
%F G.f. A(x) = Sum_{n>=1} a(n) * x^(6*n-5) satisfies:
%F (1) A( A(x) + A(x)^4 ) = G(x),
%F (2) G( A(x) - A(x)^4 ) = A(x),
%F (3) A( A(x) - A(x)^4 ) = -G(-x),
%F (4) A( A(x-x^4) + A(x-x^4)^4 ) = x,
%F where G(x) = x + G(x)^4 = Sum_{n>=1} binomial(4*n-3,n-1)/(4*n-3) * x^(3*n-2).
%e G.f.: A(x) = x + 2*x^7 + 35*x^13 + 812*x^19 + 21359*x^25 + 623244*x^31 + 18568947*x^37 + 638475040*x^43 + 13249877870*x^49 + 2024051330358*x^55 +...
%e such that Series_Reversion( A(x) + A(x)^4 ) = A(x) - A(x)^4, where
%e A(x)^4 = x^4 + 8*x^10 + 164*x^16 + 4120*x^22 + 113970*x^28 + 3416128*x^34 + 104776764*x^40 + 3565389600*x^46 + 88390775151*x^52 +...
%e A(x) + A(x)^4 = x + x^4 + 2*x^7 + 8*x^10 + 35*x^13 + 164*x^16 + 812*x^19 + 4120*x^22 + 21359*x^25 + 113970*x^28 + 623244*x^31 + 3416128*x^34 + 18568947*x^37 + 104776764*x^40 + 638475040*x^43 + 3565389600*x^46 + 13249877870*x^49 + 88390775151*x^52 + 2024051330358*x^55 +...
%e Also,
%e A( A(x) + A(x)^4 ) = x + x^4 + 4*x^7 + 22*x^10 + 140*x^13 + 969*x^16 + 7084*x^19 + 53820*x^22 +...+ binomial(4*n-3,n-1)/(4*n-3)*x^(3*n-2) +...
%e which is a g.f. of A002293.
%e Note that the following is an integer series:
%e sqrt(A(x)/x) = 1 + x^6 + 17*x^12 + 389*x^18 + 10146*x^24 + 294863*x^30 + 8741468*x^36 + 301536587*x^42 + 6008625027*x^48 + 994498807123*x^54 - 179176440388960*x^60 + 65367342797524884*x^66 - 27123073712119583646*x^72 +...
%o (PARI) {a(n) = my(Oxn=x*O(x^(6*n)), A = x +Oxn); for(i=1, 6*n, A = A + (x - subst(A + A^4, x, A - A^4 ))/2); polcoeff(A, 6*n-5)}
%o for(n=1,25,print1(a(n),", "))
%Y Cf. A277292, A277293, A002293.
%K sign
%O 1,2
%A _Paul D. Hanna_, Oct 12 2016
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