A295542 - OEIS

%I #6 Nov 24 2017 18:55:57

%S 1,1,-11,223,-5625,158762,-4799418,151973098,-4975847295,167086255160,

%T -5722690768355,199141613242149,-7020917669339366,250245906882513529,

%U -9002510253230516540,326450790268343159148,-11919962635374125266108,437889823386052090129335,-16172675073048385598095916,600164831304018730909401880,-22367394106767661737796716563

%N G.f. A(x) satisfies: x = 1 - A(x)^2 - A(x)^3 + A(x)^6.

%H Paul D. Hanna, <a href="/A295542/b295542.txt">Table of n, a(n) for n = 0..500</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = 1 + Series_Reversion( x + 11*x^2 + 19*x^3 + 15*x^4 + 6*x^5 + x^6 ).

%F (2) 1/A(x) = 1 - Series_Reversion( x*(1 + 6*x - 15*x^2 + 14*x^3 - 6*x^4 + x^5)/(1-x)^6 ).

%F (3) 1 + x = A( x + 11*x^2 + 19*x^3 + 15*x^4 + 6*x^5 + x^6 ).

%F (4) 1/(1-x) = A ( x*(1 + 6*x - 15*x^2 + 14*x^3 - 6*x^4 + x^5)/(1-x)^6 ).

%F (5) x = (1 - A(x)) * (1 + A(x) - A(x)^3 - A(x)^4 - A(x)^5).

%e G.f.: A(x) = 1 + x - 11*x^2 + 223*x^3 - 5625*x^4 + 158762*x^5 - 4799418*x^6 + 151973098*x^7 - 4975847295*x^8 + 167086255160*x^9 - 5722690768355*x^10 +...

%e such that x = 1 - A(x)^2 - A(x)^3 + A(x)^6.

%e RELATED SERIES.

%e x/(1 - A(x)) = -1 - 11*x + 102*x^2 - 2050*x^3 + 51591*x^4 - 1454635*x^5 + 43948748*x^6 - 1391121273*x^7 + 45535954995*x^8 +...

%e which equals 1 + A(x) - A(x)^3 - A(x)^4 - A(x)^5.

%e 1/A(x) = 1 - x + 12*x^2 - 246*x^3 + 6226*x^4 - 175995*x^5 + 5325019*x^6 - 168710772*x^7 + 5526022977*x^8 - 185614181252*x^9 +...

%e which equals 1 - Series_Reversion( x*(1 + 6*x - 15*x^2 + 14*x^3 - 6*x^4 + x^5)/(1-x)^6 ).

%e A(x)^2 = 1 + 2*x - 21*x^2 + 424*x^3 - 10683*x^4 + 301368*x^5 - 9107833*x^6 + 288345846*x^7 - 9439712721*x^8 + 316950794646*x^9 +...

%e A(x)^3 = 1 + 3*x - 30*x^2 + 604*x^3 - 15207*x^4 + 428850*x^5 - 12958169*x^6 + 410195916*x^7 - 13427682006*x^8 + 450825153694*x^9 +...

%e A(x)^4 = 1 + 4*x - 38*x^2 + 764*x^3 - 19229*x^4 + 542196*x^5 - 16381732*x^6 + 518543720*x^7 - 16973826519*x^8 + 569869741804*x^9 +...

%e A(x)^5 = 1 + 5*x - 45*x^2 + 905*x^3 - 22780*x^4 + 642351*x^5 - 19408265*x^6 + 614354735*x^7 - 20110293765*x^8 + 675177254035*x^9 +...

%e A(x)^6 = 1 + 6*x - 51*x^2 + 1028*x^3 - 25890*x^4 + 730218*x^5 - 22066002*x^6 + 698541762*x^7 - 22867394727*x^8 + 767775948340*x^9 +...

%e where A(x)^2 = 1 - x - A(x)^3 + A(x)^6.

%o (PARI) {a(n) = my(A=1); A = 1 - serreverse( subst(1 - x^2 - x^3 + x^6, x, 1-x) +x^2*O(x^n) ); polcoeff(A, n)}

%o for(n=0, 40, print1(a(n), ", "))

%o (PARI) {a(n) = my(A=1); A = 1/(1 - serreverse( x*(1 + 6*x - 15*x^2 + 14*x^3 - 6*x^4 + x^5)/(1-x)^6 +x^2*O(x^n)) ); polcoeff(A, n)}

%o for(n=0, 40, print1(a(n), ", "))

%Y Cf. A295543, A295544.

%K sign

%O 0,3

%A _Paul D. Hanna_, Nov 24 2017