OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..375
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1 + 3*A(x) = Sum_{n=-oo..+oo} (x + A(x)^n)^n.
(2) 1 + 3*A(x) = Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + x*A(x)^n)^n.
a(n) ~ c * d^n / n^(3/2), where d = 4.3567990762147483733716316... and c = 0.1090197875830619950924... - Vaclav Kotesovec, Jan 22 2025
EXAMPLE
G.f: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 18*x^5 + 55*x^6 + 187*x^7 + 659*x^8 + 2411*x^9 + 8888*x^10 + 33416*x^11 + 127319*x^12 + ...
where 1 + 3*A(x) = Sum_{n=-oo..+oo} (x + A(x)^n)^n.
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.2167316111650431698662382153716979000073779079648506...
where 2 = Sum_{n=-oo..+oo} (t + 1/3^n)^n,
also, 2 = Sum_{n=-oo..+oo} (1/3)^(n^2) / (1 + t/3^n)^n.
A(t) = 1/4 at t = 0.185848695796693106893457288965586148176377244454644...
where 7/4 = Sum_{n=-oo..+oo} (t + 1/4^n)^n,
also, 7/4 = Sum_{n=-oo..+oo} (1/4)^(n^2) / (1 + t/4^n)^n.
A(t) = 1/5 at t = 0.159367217342154668563892986702369800216583530179904...
where 8/5 = Sum_{n=-oo..+oo} (t + 1/5^n)^n,
also, 8/5 = Sum_{n=-oo..+oo} 1/(5^n + t)^n.
A(1/5) = 0.28267664469643998678469927594507868991939755342415...
where 1 + 3*A(1/5) = Sum_{n=-oo..+oo} (1/5 + A(1/5)^n)^n.
A(1/6) = 0.21276190263087786208514906090343419861687760049670...
A(1/7) = 0.17318736394146169448330543884000299105132488182099...
A(1/8) = 0.14667325772966314898436041245607358265234843090235...
A(1/10) = 0.1127641318058543678973455784350429686984190649814...
PROG
(PARI) {a(n) = my(V=[0, 1], A=x); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(n=-#V, #V, (x + A^n)^n ) - 3*A, #V-1) ); V[n+1]}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 10 2025
STATUS
approved