A295542 G.f. A(x) satisfies: x = 1 - A(x)^2 - A(x)^3 + A(x)^6.

1, 1, -11, 223, -5625, 158762, -4799418, 151973098, -4975847295, 167086255160, -5722690768355, 199141613242149, -7020917669339366, 250245906882513529, -9002510253230516540, 326450790268343159148, -11919962635374125266108, 437889823386052090129335, -16172675073048385598095916, 600164831304018730909401880, -22367394106767661737796716563

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

G.f. A(x) satisfies:

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

(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 ).

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

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

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

Example

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 +...
such that x = 1 - A(x)^2 - A(x)^3 + A(x)^6.
RELATED SERIES.
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 +...
which equals 1 + A(x) - A(x)^3 - A(x)^4 - A(x)^5.
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 +...
which equals 1 - Series_Reversion( x*(1 + 6*x - 15*x^2 + 14*x^3 - 6*x^4 + x^5)/(1-x)^6 ).
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 +...
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 +...
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 +...
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 +...
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 +...
where A(x)^2 = 1 - x - A(x)^3 + A(x)^6.

Prog

(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)}
for(n=0, 40, print1(a(n), ", "))
(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)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A295543, A295544.

Sequence in context: A308438 A103611 A142541 * A089917 A294388 A281257

Adjacent sequences: A295539 A295540 A295541 * A295543 A295544 A295545

Keyword

sign

Author

Paul D. Hanna, Nov 24 2017

Status

approved