OFFSET
0,3
COMMENTS
a(n) ~ c*d^n, where d = 3.354756161514554082... and c = 0.322808122437862772...
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 14*x^3 + 44*x^4 + 130*x^5 + 496*x^6 + 1586*x^7 + 5128*x^8 + 17764*x^9 + 59492*x^10 + 196368*x^11 + 659330*x^12 + ...
The coefficients in 1/A(x)^n begin
n = 1: [1, -1, -3, -7, -11, -5, -87, -131, -111, ...];
n = 2: [1, -2, -5, -8, 1, 54, -49, 96, 753, ...];
n = 3: [1, -3, -6, -4, 27, 129, -72, 132, 1041, ...];
n = 4: [1, -4, -6, 4, 59, 184, -260, -168, 749, ...];
n = 5: [1, -5, -5, 15, 90, 194, -650, -670, 675, ...];
n = 6: [1, -6, -3, 28, 114, 144, -1226, -1068, 1851, ...];
n = 7: [1, -7, 0, 42, 126, 28, -1932, -974, 5131, ...];
n = 8: [1, -8, 4, 56, 122, -152, -2684, 8, 10915, ...];
n = 9: [1, -9, 9, 69, 99, -387, -3381, 2223, 18990, ...];
...
The unsigned coefficients form the series abs(1/A(x)^n) and begin
n = 1: [1, 1, 3, 7, 11, 5, 87, 131, 111, ...];
n = 2: [1, 2, 5, 8, 1, 54, 49, 96, 753, ...];
n = 3: [1, 3, 6, 4, 27, 129, 72, 132, 1041, ...];
n = 4: [1, 4, 6, 4, 59, 184, 260, 168, 749, ...];
n = 5: [1, 5, 5, 15, 90, 194, 650, 670, 675, ...];
n = 6: [1, 6, 3, 28, 114, 144, 1226, 1068, 1851, ...];
n = 7: [1, 7, 0, 42, 126, 28, 1932, 974, 5131, ...];
n = 8: [1, 8, 4, 56, 122, 152, 2684, 8, 10915, ...];
n = 9: [1, 9, 9, 69, 99, 387, 3381, 2223, 18990, ...];
...
The g.f. A(x) causes the following sum to equal x:
x = Sum_{n>=1} -(-1)^(n mod 3) * x^n * abs(1/A(x)^n), i.e.,
x = x *(1 + 1*x + 3*x^2 + 7*x^3 + 11*x^4 + 5*x^5 + ...)
- x^2*(1 + 2*x + 5*x^2 + 8*x^3 + 1*x^4 + 54*x^5 + ...)
- x^3*(1 + 3*x + 6*x^2 + 4*x^3 + 27*x^4 + 129*x^5 + ...)
+ x^4*(1 + 4*x + 6*x^2 + 4*x^3 + 59*x^4 + 184*x^5 + ...)
- x^5*(1 + 5*x + 5*x^2 + 15*x^3 + 90*x^4 + 194*x^5 + ...)
- x^6*(1 + 6*x + 3*x^2 + 28*x^3 + 114*x^4 + 144*x^5 + ...)
+ x^7*(1 + 7*x + 0*x^2 + 42*x^3 + 126*x^4 + 28*x^5 + ...)
- x^8*(1 + 8*x + 4*x^2 + 56*x^3 + 122*x^4 + 152*x^5 + ...)
- x^9*(1 + 9*x + 9*x^2 + 69*x^3 + 99*x^4 + 387*x^5 + ...)
+ ...
SPECIFIC VALUES.
A(t) = 5/2 at t = 0.2439659541767366435324668966...
A(t) = 9/4 at t = 0.2353840551855704278278595775153205...
A(t) = 2 at t = 0.223632028538443643064216621211748830622323...
A(t) = 7/4 at t = 0.20657376416781114780239812337080772813821570967375...
A(t) = 5/3 at t = 0.19903893798837337323032062934473873273603265809256...
A(t) = 3/2 at t = 0.17957316117018144667020305457516566265734157029382...
A(t) = 4/3 at t = 0.15062899618726958291222230561587359930435166593485...
A(t) = 5/4 at t = 0.13003050002674430753834725398705188137096165805154...
A(1/4) = 2.7283787868137581335275349...
A(1/5) = 1.6765917121780458577176419255019373572188147733104...
A(1/6) = 1.4165864667091498858294378983473697621659718923072...
A(1/7) = 1.2992396855197346868111927874895540857809847001779...
A(1/8) = 1.2327343901263073476621403289881692603672322742418...
PROG
(PARI) {a(n) = my(V=[1]); for(i=0, n, V=concat(V, 't); A=Ser(V);
V[#V] = 't - polcoef(-x + sum(m=1, #A+2, -(-1)^(m%3) * x^m * Ser(abs(Vec(1/A^m))) ), #V)); polcoef(H=A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 02 2025
STATUS
approved