A277177 - OEIS

%I #10 Oct 16 2016 17:55:41

%S 1,3,-24,784,-46896,4317456,-562745520,98690557584,-22460740839216,

%T 6450171240032784,-2284564885046722992,979187134208612772240,

%U -499793632523433181810736,299667123722863936449037584,-208593901698504257051080848048,166857381492340592665576625921168,-152017413229327953521998890673326384,156505079591897717881208227590259134480

%N G.f. A(x) satisfies: Series_Reversion( A(x)/(1+x)^3 ) = A(x)/(1-x)^3.

%C Compare to: Series_Reversion( F(x)*(1+x) ) = F(x)*(1-x) when F(x) = x/(1-x^2).

%H Paul D. Hanna, <a href="/A277177/b277177.txt">Table of n, a(n) for n = 1..151</a>

%F G.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1)  also satisfies:

%F (1) A( A(x)/(1-x)^3 ) = x*(1 + A(x)/(1-x)^3 )^3.

%F (2) A( A(x)/(1+x)^3 ) = x*(1 - A(x)/(1+x)^3 )^3.

%e G.f.: A(x) = x + 3*x^3 - 24*x^5 + 784*x^7 - 46896*x^9 + 4317456*x^11 - 562745520*x^13 + 98690557584*x^15 - 22460740839216*x^17 + 6450171240032784*x^19 - 2284564885046722992*x^21 + 979187134208612772240*x^23 - 499793632523433181810736*x^25 +...

%e such that Series_Reversion( A(x)/(1+x)^3 ) = A(x)/(1-x)^3.

%e RELATED SERIES.

%e A(x)/(1-x)^3 = x + 3*x^2 + 9*x^3 + 19*x^4 + 9*x^5 - 21*x^6 + 713*x^7 + 2211*x^8 - 42423*x^9 - 133189*x^10 + 4047369*x^11 + 12499251*x^12 - 537523063*x^13 - 1646019573*x^14 + 95377567305*x^15 + 290533237571*x^16 - 21876919847991*x^17 - 66406981689381*x^18 + 6316871587746185*x^19 + 19127958788458707*x^20 +...

%e A( A(x)/(1-x)^3 ) = x + 3*x^2 + 12*x^3 + 46*x^4 + 147*x^5 + 357*x^6 + 822*x^7 + 4056*x^8 + 14469*x^9 - 93897*x^10 - 549840*x^11 + 10758642*x^12 + 56601703*x^13 - 1469357319*x^14 - 7645881510*x^15 + 266606239932*x^16 + 1374106441785*x^17 - 62103211575765*x^18 - 317982458976204*x^19 + 18131631145035702*x^20 +...

%e Series_Reversion(A(x)) = x - 3*x^3 + 51*x^5 - 1684*x^7 + 89631*x^9 - 7218867*x^11 + 842669332*x^13 - 136419363048*x^15 + 29359346719959*x^17 - 8103779638873081*x^19 + 2787880624127169867*x^21 - 1168604294464477079760*x^23 + 586059363388037097318788*x^25 +...

%e Incidentally,

%e (A(x)/x)^(1/3) = 1 + x^2 - 9*x^4 + 279*x^6 - 16262*x^8 + 1476338*x^10 - 190883546*x^12 + 33312398406*x^14 - 7557834013343*x^16 + 2165888855602865*x^18 - 766008084752723233*x^20 + 327970244729039892575*x^22 - 167268383573932822246336*x^24 + 100229670800696383970592920*x^26 - 69734815267693299184321137784*x^28 + 55760405033094966392326154518792*x^30 +...

%e appears to be an integer series.

%o (PARI) {a(n) = my(A = x +x*O(x^(2*n)));for(i=1,2*n, A = A + (x - subst(A/(1+x +x*O(x^(2*n)))^3,x, A/(1-x +x*O(x^(2*n)))^3))/2); polcoeff(A, 2*n-1)}

%o for(n=1, 20, print1(a(n), ", "))

%Y Cf. A277038.

%K sign

%O 1,2

%A _Paul D. Hanna_, Oct 02 2016