A071878 - OEIS

%I #17 Jul 15 2024 06:11:34

%S 1,6,47,420,4058,41286,435739,4726644,52373294,590247900,6744908118,

%T 77969430864,910131055980,10712886629958,127012431301779,

%U 1515405441505380,18181513435560278,219219809605566132,2654917102081791394,32281268283914386200

%N G.f. D(x) satisfies: D(x) = (1 + x*D(x))*(1 + 2*x*D(x))*(1 + 3*x*D(x)).

%H Paul D. Hanna, <a href="/A071878/b071878.txt">Table of n, a(n) for n = 0..520</a>

%F From _Paul D. Hanna_, Mar 01 2021: (Start)

%F G.f.: D(x) = (1/x) * Series_Reversion( x / ((1 + x)*(1 + 2*x)*(1 + 3*x)) ).

%F G.f. D = D(x) and related functions A = A(x), B = B(x), C = C(x), satisfy:

%F (1.a) A = 1/((1 - 2*x*B)*(1 - 3*x*C)).

%F (1.b) B = 1/((1 - x*A)*(1 - 3*x*C)).

%F (1.c) C = 1/((1 - x*A)*(1 - 2*x*B)).

%F (1.d) D = 1/((1 - x*A)*(1 - 2*x*B)*(1 - 3*x*C)).

%F (1.e) D = sqrt(A*B*C).

%F (2.a) A = (1 + 2*x*D)*(1 + 3*x*D).

%F (2.b) B = (1 + x*D)*(1 + 3*x*D).

%F (2.c) C = (1 + x*D)*(1 + 2*x*D).

%F (2.d) D = (sqrt(24*A + 1) - 5)/(12*x) = (sqrt(12*B + 4) - 4)/(6*x) = (sqrt(8*C + 1) - 3)/(4*x).

%F (3.a) A = B/(1 - x*B) = C/(1 - 2*x*C) = D/(1 + x*D).

%F (3.b) B = C/(1 - x*C) = A/(1 + x*A) = D/(1 + 2*x*D).

%F (3.c) C = A/(1 + 2*x*A) = B/(1 + x*B) = D/(1 + 3*x*D).

%F (3.d) D = A/(1 - x*A) = B/(1 - 2*x*B) = C/(1 - 3*x*C).

%F (3.e) 1 = (1 + x*A)*(1 - x*B) = (1 + 2*x*A)*(1 - 2*x*C) = (1 + x*B)*(1 - x*C).

%F (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).

%F (4.a) A = (1 + x*A)*(1 + 2*x*A)/(1 - x*A)^2.

%F (4.b) B = (1 - x^2*B^2)/(1 - 2*x*B)^2.

%F (4.c) C = (1 - x*C)*(1 - 2*x*C)/(1 - 3*x*C)^2.

%F (4.d) D = (1 + x*D)*(1 + 2*x*D)*(1 + 3*x*D).

%F (5.a) A = (1/x)*Series_Reversion( x*(1 - x)^2 / ((1 + x)*(1 + 2*x)) ).

%F (5.b) B = (1/x)*Series_Reversion( x*(1 - 2*x)^2 / (1 - x^2) ).

%F (5.c) C = (1/x)*Series_Reversion( x*(1 - 3*x)^2 / ((1 - x)*(1 - 2*x)) ).

%F (5.d) D = (1/x)*Series_Reversion( x / ((1 + x)*(1 + 2*x)*(1 + 3*x)) ).

%F (End)

%F 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

%e 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 + ...

%e such that D(x) = (1 + x*D(x))*(1 + 2*x*D(x))*(1 + 3*x*D(x))

%e and also D(x) = sqrt(A(x)*B(x)*C(x)) where

%e 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 + ...

%e 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 + ...

%e 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 + ...

%e RELATED SERIES.

%e 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 + ...

%e 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 + ...

%e 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 + ...

%e 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 + ...

%t CoefficientList[1/x * InverseSeries[Series[x/((1 + x)*(1 + 2*x)*(1 + 3*x)), {x, 0, 20}], x], x] (* _Vaclav Kotesovec_, Mar 02 2021 *)

%o (PARI) {d(n) = my(A=1, B=1, C=1, D=1); for(i=1, n,

%o A = 1/((1-2*x*B)*(1-3*x*C) +x*O(x^n));

%o B = 1/((1-1*x*A)*(1-3*x*C) +x*O(x^n));

%o C = 1/((1-1*x*A)*(1-2*x*B) +x*O(x^n));

%o D = sqrt(A*B*C)); polcoeff(D, n)}

%o for(n=0, 30, print1(c(n), ", ")) \\ _Paul D. Hanna_, Mar 01 2021

%o (PARI) /* By Series Reversion: */

%o {d(n) = my(D = 1/x*serreverse( x/((1 + x)*(1 + 2*x)*(1 + 3*x) +x*O(x^n)))); polcoeff(D,n)}

%o for(n=0,11, print1(d(n),", ")) \\ _Paul D. Hanna_, Mar 01 2021

%Y Cf. A341961 (A(x)), A341962 (B(x)), A341963 (C(x)).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 10 2002

%E Entry revised by _Paul D. Hanna_, Mar 01 2021