A295541 G.f. A(x) satisfies: x = 1 + 2*A(x) - 6*A(x)^2 + 3*A(x)^3.

1, -1, 3, -21, 180, -1728, 17766, -191322, 2130381, -24328755, 283378095, -3353616513, 40209584220, -487400777388, 5963013401400, -73536394960872, 913148848213740, -11408099402698488, 143288658075653370, -1808347110471143550, 22919638640740373070, -291612557490911415990, 3723227753617063771380, -47688275868038913956940, 612581097400862637697650, -7889900533870136800820958

Offset

0,3

Links

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

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Formula

G.f. A(x) satisfies:

(1) A(x) = 1 - Series_Reversion( x + 3*x^2 - 3*x^3 ).

(2) 1/A(x) = 1 + Series_Reversion( x*(1 + 5*x + x^2)/(1+x)^3 ).

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

(4) 1/(1+x) = A ( x*(1 + 5*x + x^2)/(1+x)^3 ).

(5) x = (1 - A(x)) * (1 + 3*A(x) - 3*A(x)^2).

Example

G.f.: A(x) = 1 - x + 3*x^2 - 21*x^3 + 180*x^4 - 1728*x^5 + 17766*x^6 - 191322*x^7 + 2130381*x^8 - 24328755*x^9 + 283378095*x^10 +...
such that x = (1 - A(x)) * (1 + 3*A(x) - 3*A(x)^2).
RELATED SERIES.
x/(1 - A(x)) = 1 + 3*x - 12*x^2 + 81*x^3 - 693*x^4 + 6642*x^5 - 68229*x^6 + 734346*x^7 - 8173791*x^8 + 93317103*x^9 - 1086705477*x^10 +...
which equals 1 + 3*A(x) - 3*A(x)^2.
1/A(x) = 1 + x - 2*x^2 + 16*x^3 - 137*x^4 + 1321*x^5 - 13610*x^6 + 146770*x^7 - 1635878*x^8 + 18694960*x^9 - 217876232*x^10 +...
which equals 1 - Series_Reversion( x*(1 + 5*x + x^2)/(1+x)^3 ).
A(x)^2 = 1 - 2*x + 7*x^2 - 48*x^3 + 411*x^4 - 3942*x^5 + 40509*x^6 - 436104*x^7 + 4854978*x^8 - 55434456*x^9 + 645613254*x^10 +...
A(x)^3 = 1 - 3*x + 12*x^2 - 82*x^3 + 702*x^4 - 6732*x^5 + 69174*x^6 - 744660*x^7 + 8289702*x^8 - 94649742*x^9 + 1102307778*x^10 +...
where x = 1 + 2*A(x) - 6*A(x)^2 + 3*A(x)^3.

Prog

(PARI) {a(n) = my(A=1); A = 1 - serreverse( subst( (1-x)*(1 + 3*x - 3*x^2), 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 + 5*x + x^2)/(1+x)^3 +x^2*O(x^n)) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Sequence in context: A372108 A195105 A285272 * A369941 A192946 A216171

Adjacent sequences: A295538 A295539 A295540 * A295542 A295543 A295544

Keyword

sign

Author

Paul D. Hanna, Nov 24 2017

Status

approved