A275754 G.f. A(x) satisfies: 1 = ...(((((A(x) - x)^(1/2) - x^2)^(1/3) - x^3)^(1/4) - x^4)^(1/5) - x^5)^(1/6) -...- x^n)^(1/(n+1)) -..., an infinite series of nested n-th roots.

1, 1, 2, 6, 25, 126, 759, 5280, 41922, 374348, 3718632, 40687982, 486272963, 6302207628, 88028389880, 1318140365566, 21062423444982, 357696122160256, 6433377303116791, 122156160094379094, 2441860821816085218, 51257318297766697640, 1127262952173081139014, 25919345794724305427250, 621907163139705123859022, 15544308709055844154008420, 404076225273612078440996928, 10908274705160761799960317228

Offset

0,3

Comments

Limit a(n)/n! = 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..520

Formula

a(n) ~ n! * (1 + 1/n^2 + 6/n^3 + 39/n^4 + 295/n^5 + 2601/n^6 + 26384/n^7 + 303669/n^8 + 3918008/n^9 + 56085221/n^10 + ...), for coefficients see A289996. - Vaclav Kotesovec, Sep 03 2017

Example

G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 126*x^5 + 759*x^6 + 5280*x^7 + 41922*x^8 + 374348*x^9 + 3718632*x^10 +...
Illustration of definition.
Start with R1 = A(x), and proceed as follows:
R2 = (R1 - x^1)^(1/2) = 1 + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 363*x^6 + 2544*x^7 +...
R3 = (R2 - x^2)^(1/3) = 1 + x^3 + 4*x^4 + 20*x^5 + 120*x^6 + 840*x^7 +...
R4 = (R3 - x^3)^(1/4) = 1 + x^4 + 5*x^5 + 30*x^6 + 210*x^7 + 1680*x^8 +...
R5 = (R4 - x^4)^(1/5) = 1 + x^5 + 6*x^6 + 42*x^7 + 336*x^8 + 3024*x^9 +...
R6 = (R5 - x^5)^(1/6) = 1 + x^6 + 7*x^7 + 56*x^8 + 504*x^9 + 5040*x^10 +...
R7 = (R6 - x^6)^(1/7) = 1 + x^7 + 8*x^8 + 72*x^9 + 720*x^10 + 7920*x^11 +...
R8 = (R7 - x^7)^(1/8) = 1 + x^8 + 9*x^9 + 90*x^10 + 990*x^11 + 11880*x^12 +...
R9 = (R8 - x^8)^(1/9) = 1 + x^9 + 10*x^10 + 110*x^11 + 1320*x^12 + 17160*x^13 +...
etc., to approach the value 1 as a limit.
Generating Method.
The g.f. may be attained as a limit of the following process.
Start with 1, add x^n and raise that result to the n power, add x^(n-1) and raise that result to the (n-1) power, and continue until you reach x^1 to approximate the g.f. A(x).
For example, say n = 6, then start with S = 1 and work backwards like so:
S = (S + x^6)^6 = 1 + 6*x^6 + 15*x^12 +...
S = (S + x^5)^5 = 1 + 5*x^5 + 30*x^6 + 10*x^10 + 120*x^11 +...
S = (S + x^4)^4 = 1 + 4*x^4 + 20*x^5 + 120*x^6 + 6*x^8 + 60*x^9 + 550*x^10 +...
S = (S + x^3)^3 = 1 + 3*x^3 + 12*x^4 + 60*x^5 + 363*x^6 + 24*x^7 + 186*x^8 +...
S = (S + x^2)^2 = 1 + 2*x^2 + 6*x^3 + 25*x^4 + 126*x^5 + 759*x^6 + 240*x^7 +...
S = (S + x)^1 = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 126*x^5 + 759*x^6 + 240*x^7 +...
which agrees with A(x) up to the coefficient of x^6.
RELATED SERIES.
Log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 75*x^4/4 + 486*x^5/5 + 3639*x^6/6 + 30437*x^7/7 + 283675*x^8/8 + 2913520*x^9/9 + 32744938*x^10/10 + 399922799*x^11/11 + 5276272191*x^12/12 + 74800085777*x^13/13 + 1134180192743*x^14/14 +...

Prog

(PARI) {a(n) = my(A=1); for(k=1, n, A = A^(n+2-k) + x^(n+1-k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A); if( n<1, n==0, A = n + O(x); n--; forstep(k=n, 1, -1, A = ((1 + x^k * (1 + x * A))^k - 1) / x^k); polcoeff(A, n))}; /* Michael Somos, May 16 2018 */

Crossrefs

Cf. A275753, A289996.

Sequence in context: A188687 A030859 A030877 * A030883 A030899 A030907

Adjacent sequences: A275751 A275752 A275753 * A275755 A275756 A275757

Keyword

nonn

Author

Paul D. Hanna, Aug 16 2016

Status

approved