OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..520
FORMULA
From Paul D. Hanna, Mar 01 2021: (Start)
G.f.: D(x) = (1/x) * Series_Reversion( x / ((1 + x)*(1 + 2*x)*(1 + 3*x)) ).
G.f. D = D(x) and related functions A = A(x), B = B(x), C = C(x), satisfy:
(1.a) A = 1/((1 - 2*x*B)*(1 - 3*x*C)).
(1.b) B = 1/((1 - x*A)*(1 - 3*x*C)).
(1.c) C = 1/((1 - x*A)*(1 - 2*x*B)).
(1.d) D = 1/((1 - x*A)*(1 - 2*x*B)*(1 - 3*x*C)).
(1.e) D = sqrt(A*B*C).
(2.a) A = (1 + 2*x*D)*(1 + 3*x*D).
(2.b) B = (1 + x*D)*(1 + 3*x*D).
(2.c) C = (1 + x*D)*(1 + 2*x*D).
(2.d) D = (sqrt(24*A + 1) - 5)/(12*x) = (sqrt(12*B + 4) - 4)/(6*x) = (sqrt(8*C + 1) - 3)/(4*x).
(3.a) A = B/(1 - x*B) = C/(1 - 2*x*C) = D/(1 + x*D).
(3.b) B = C/(1 - x*C) = A/(1 + x*A) = D/(1 + 2*x*D).
(3.c) C = A/(1 + 2*x*A) = B/(1 + x*B) = D/(1 + 3*x*D).
(3.d) D = A/(1 - x*A) = B/(1 - 2*x*B) = C/(1 - 3*x*C).
(3.e) 1 = (1 + x*A)*(1 - x*B) = (1 + 2*x*A)*(1 - 2*x*C) = (1 + x*B)*(1 - x*C).
(3.f) 1 = (1 - x*A)*(1 + x*D) = (1 - 2*x*B)*(1 + 2*x*D) = (1 - 3*x*C)*(1 + 3*x*D).
(4.a) A = (1 + x*A)*(1 + 2*x*A)/(1 - x*A)^2.
(4.b) B = (1 - x^2*B^2)/(1 - 2*x*B)^2.
(4.c) C = (1 - x*C)*(1 - 2*x*C)/(1 - 3*x*C)^2.
(4.d) D = (1 + x*D)*(1 + 2*x*D)*(1 + 3*x*D).
(5.a) A = (1/x)*Series_Reversion( x*(1 - x)^2 / ((1 + x)*(1 + 2*x)) ).
(5.b) B = (1/x)*Series_Reversion( x*(1 - 2*x)^2 / (1 - x^2) ).
(5.c) C = (1/x)*Series_Reversion( x*(1 - 3*x)^2 / ((1 - x)*(1 - 2*x)) ).
(5.d) D = (1/x)*Series_Reversion( x / ((1 + x)*(1 + 2*x)*(1 + 3*x)) ).
(End)
a(n) ~ sqrt((3*s + 11*r*s^2 + 9*r^2*s^3)/(Pi*(22 + 36*r*s))) / (n^(3/2)*r^(n + 1/2)), where r = 4 - sqrt(503/3) * cos(arctan(359*sqrt(359/3)/5196)/3)/2 + sqrt(503) * sin(arctan(359*sqrt(359/3)/5196)/3)/2 = 0.07627811703169412709742160523783922642030319519275992338... and s = 3.4807233253858558164460728604043678335213362043902693560668... are positive real roots of the system of equations (1 + r*s)*(1 + 2*r*s)*(1 + 3*r*s) = s, 2*r*(3 + 11*r*s + 9*r^2*s^2) = 1. - Vaclav Kotesovec, Mar 02 2021
EXAMPLE
G.f. D(x) 1 + 6*x + 47*x^2 + 420*x^3 + 4058*x^4 + 41286*x^5 + 435739*x^6 + 4726644*x^7 + 52373294*x^8 + 590247900*x^9 + 6744908118*x^10 + ...
such that D(x) = (1 + x*D(x))*(1 + 2*x*D(x))*(1 + 3*x*D(x))
and also D(x) = sqrt(A(x)*B(x)*C(x)) where
A(x) = 1 + 5*x + 36*x^2 + 307*x^3 + 2880*x^4 + 28714*x^5 + 298620*x^6 + 3203183*x^7 + 35181792*x^8 + 393697030*x^9 + 4472679816*x^10 + ...
B(x) = 1 + 4*x + 27*x^2 + 224*x^3 + 2070*x^4 + 20444*x^5 + 211239*x^6 + 2255200*x^7 + 24680862*x^8 + 275408456*x^9 + 3121711758*x^10 + ...
C(x) = 1 + 3*x + 20*x^2 + 165*x^3 + 1520*x^4 + 14982*x^5 + 154588*x^6 + 1648713*x^7 + 18029456*x^8 + 201063402*x^9 + 2277890472*x^10 + ...
RELATED SERIES.
D(x)^2 = A(x)*B(x)*C(x) = 1 + 12*x + 130*x^2 + 1404*x^3 + 15365*x^4 + 170748*x^5 + 1924762*x^6 + 21971760*x^7 + 253573386*x^8 + 2954377800*x^9 + ...
B(x)*C(x) = D(x) + x*D(x)^2 = 1 + 7*x + 59*x^2 + 550*x^3 + 5462*x^4 + 56651*x^5 + 606487*x^6 + 6651406*x^7 + 74345054*x^8 + ...
A(x)*C(x) = D(x) + 2*x*D(x)^2 = 1 + 8*x + 71*x^2 + 680*x^3 + 6866*x^4 + 72016*x^5 + 777235*x^6 + 8576168*x^7 + 96316814*x^8 + ...
A(x)*B(x) = D(x) + 3*x*D(x)^2 = 1 + 9*x + 83*x^2 + 810*x^3 + 8270*x^4 + 87381*x^5 + 947983*x^6 + 10500930*x^7 + 118288574*x^8 + ...
MATHEMATICA
CoefficientList[1/x * InverseSeries[Series[x/((1 + x)*(1 + 2*x)*(1 + 3*x)), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Mar 02 2021 *)
PROG
(PARI) {d(n) = my(A=1, B=1, C=1, D=1); for(i=1, n,
A = 1/((1-2*x*B)*(1-3*x*C) +x*O(x^n));
B = 1/((1-1*x*A)*(1-3*x*C) +x*O(x^n));
C = 1/((1-1*x*A)*(1-2*x*B) +x*O(x^n));
D = sqrt(A*B*C)); polcoeff(D, n)}
for(n=0, 30, print1(c(n), ", ")) \\ Paul D. Hanna, Mar 01 2021
(PARI) /* By Series Reversion: */
{d(n) = my(D = 1/x*serreverse( x/((1 + x)*(1 + 2*x)*(1 + 3*x) +x*O(x^n)))); polcoeff(D, n)}
for(n=0, 11, print1(d(n), ", ")) \\ Paul D. Hanna, Mar 01 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 10 2002
EXTENSIONS
Entry revised by Paul D. Hanna, Mar 01 2021
STATUS
approved