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a(n) ~ c * n^(n-1), where c = A072364 = exp(-exp(-1)). - Vaclav Kotesovec, Feb 27 2022

Vaclav Kotesovec, <a href="/A351917/b351917.txt">Table of n, a(n) for n = 0..380</a>

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G.f. A(x) = limit of F(n) as lim_{n approaches ->infinity, } F(n), where: F(0) = 1, F(n) = ( F(n-1)^n + (n*x)^n )^(1/n) for n = 1,2,3,...

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 + 5297*x^6 + 79803*x^7 + 1425901*x^8 + 29317826*x^9 + 682661738*x^10 + ...

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1, 1, 2, 7, 46, 428, 5297, 79803, 1425901, 29317826, 682661738, 17737351760, 508899803500, 15976473483553, 544826377810898, 20054787151757636, 792552513383685078, 33469556711571645662, 1504188914267750552104, 71681627511482293102424, 3610465782547332984625361

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The table of coefficients of x^k in A(x)^n, for k >= 0 and n >= 1, begins:

allocated for Paul D. Hanna

G.f. A(x) = limit of F(n) as n approaches infinity, where: F(0) = 1, F(n) = ( F(n-1)^n + (n*x)^n )^(1/n) for n = 1,2,3,...

1, 1, 2, 7, 46, 428, 5297, 79803, 1425901, 29317826, 682661738, 17737351760, 508899803500, 15976473483553, 544826377810898, 20054787151757636, 792552513383685078, 33469556711571645662, 1504188914267750552104

0,3

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 + 5297*x^6 + 79803*x^7 + 1425901*x^8 + 29317826*x^9 + 682661738*x^10 + ...

The g.f. A(x) equals the limit of the following process.

Start with F(0) = 1, then repeat F(n) = ( F(n-1)^n + (n*x)^n )^(1/n) for n = 1..N, for some positive integer N; the g.f. of this sequence equals the limit as N approaches infinity.

Example: start with F(0) = 1, then continue as follows.

F(1) = ( F(0)^1 + (1*x)^1 )^(1/1) = 1 + x;

F(2) = ( F(1)^2 + (2*x)^2 )^(1/2) = 1 + x + 2*x^2 - 2*x^3 + 4*x^5 - 6*x^6 - 2*x^7 + 22*x^8 + ...;

F(3) = ( F(2)^3 + (3*x)^3 )^(1/3) = 1 + x + 2*x^2 + 7*x^3 - 18*x^4 - 5*x^5 + 21*x^6 + 232*x^7 + ...;

F(4) = ( F(3)^4 + (4*x)^4 )^(1/4) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 - 197*x^5 + 21*x^6 - 216*x^7 + ...;

F(5) = ( F(4)^5 + (5*x)^5 )^(1/5) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 - 2479*x^6 + 1034*x^7 + ...;

F(6) = ( F(5)^6 + (6*x)^6 )^(1/6) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 + 5297*x^6 - 37846*x^7 + ...;

F(7) = ( F(6)^7 + (7*x)^7 )^(1/7) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 + 5297*x^6 + 79803*x^7 + ...;

...

Continue in this way to obtain the g.f. A(x) as a limit of F(n) as n approaches infinity.

Related table.

The table of coefficients of x^k in A(x)^n, for k >= 0 and n >= 1, begins:

n=1: [1, 1, 2, 7, 46, 428, 5297, 79803, ...];

n=2: [1, 2, 5, 18, 110, 976, 11683, 172556, ...];

n=3: [1, 3, 9, 34, 198, 1677, 19388, 280326, ...];

n=4: [1, 4, 14, 56, 317, 2572, 28694, 405564, ...];

n=5: [1, 5, 20, 85, 475, 3711, 39945, 551180, ...];

n=6: [1, 6, 27, 122, 681, 5154, 53558, 720630, ...];

n=7: [1, 7, 35, 168, 945, 6972, 70035, 918016, ...]; ...

in which row n is comparable with F(n)^n given in the generating process shown above.

Related series.

log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 155*x^4/4 + 1886*x^5/5 + 28908*x^6/6 + 517539*x^7/7 + 10708979*x^8/8 + 250010863*x^9/9 + 6513735398*x^10/10 + ...

(PARI) {a(n) = my(A = 1 +x*O(x^n)) ; for(m=1, n, A = (A^m + (m*x)^m )^(1/m) ); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

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nonn

Paul D. Hanna, Feb 25 2022

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