[go: up one dir, main page]

login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A341963
G.f. C(x) satisfies: C(x) = (1 - x*C(x))*(1 - 2*x*C(x)) / (1 - 3*x*C(x))^2.
5
1, 3, 20, 165, 1520, 14982, 154588, 1648713, 18029456, 201063402, 2277890472, 26143479954, 303322798896, 3551784992172, 41920546809900, 498190056106161, 5956394533009520, 71595959974620738, 864682199472132376, 10487531546082633270, 127689839838217004000
OFFSET
0,2
LINKS
FORMULA
G.f.: C(x) = (1/x) * Series_Reversion( x*(1 - 3*x)^2 / ((1 - x)*(1 - 2*x)) ).
G.f. C = C(x) and related functions A = A(x), B = B(x), D = D(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)) ).
a(n) ~ sqrt(s*(3 - 14*r*s + 15*r^2*s^2) / (Pi*(11 - 15*r*s))) / (2*n^(3/2)*r^(n + 1/2)), where r = 0.07627811703169412709742160523783922642030319519275992338... and s = 1.9374927720056356430894528816479641920545157312336620520408... are positive real roots of the system of equations (-1 + r*s)*(-1 + 2*r*s)/(1 - 3*r*s)^2 = s, -1 + 27*r^3*s^3 + r*(3 + 9*s) - r^2*s*(5 + 27*s) = 0. - Vaclav Kotesovec, Mar 02 2021
EXAMPLE
G.f. 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 + ...
such that C(x) = 1/((1 - x*A(x))*(1 - 2*x*B(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 + ...
RELATED SERIES.
D(x) = sqrt(A(x)*B(x)*C(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 + ...
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 - 3*x)^2 / ((1 - x)*(1 - 2*x)), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Mar 02 2021 *)
PROG
(PARI) {c(n) = my(A=1, B=1, C=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)); );
polcoeff(C, n)}
for(n=0, 30, print1(c(n), ", "))
(PARI) /* By Series Reversion: */
{c(n) = my(C = 1/x*serreverse( x*(1 - 3*x)^2 / ((1 - x)*(1 - 2*x) +x*O(x^n)))); polcoeff(C, n)}
for(n=0, 30, print1(c(n), ", "))
CROSSREFS
Cf. A341961 (A(x)), A341962 (B(x)), A071878 (D(x)).
Sequence in context: A336653 A012882 A063017 * A276315 A145329 A051643
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 27 2021
STATUS
approved