|
| |
|
|
A356848
|
|
Expansion of g.f. A(x) satisfying A(x) = x * Sum_{n>=0} d^n/dx^n x^(2*n-1) * A(x)^n / n!.
|
|
6
|
|
|
|
1, 1, 5, 37, 353, 4061, 54221, 820205, 13829377, 256853629, 5208050365, 114465346733, 2711004465185, 68846143222013, 1866577974450733, 53824099877628077, 1645120108520147713, 53135285623703158429, 1808560829585046118685, 64707781796679229092045, 2428043851750587122468513
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined as follows.
(1) A(x) = x * Sum_{n>=0} d^n/dx^n x^(2*n-1) * A(x)^n / n!.
(2) B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^n / n! ) where B(x - x^2*A(x)) = x, and B(x) is the g.f. of A360579.
a(n) ~ c * n! * n^(2*LambertW(1)) / LambertW(1)^n, where c = 0.421381125316... - Vaclav Kotesovec, Feb 23 2023
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + x + 5*x^2 + 37*x^3 + 353*x^4 + 4061*x^5 + 54221*x^6 + 820205*x^7 + 13829377*x^8 + 256853629*x^9 + 5208050365*x^10 + ...
such that
A(x) = x * (1/x + d/dx x*A(x)) + (d^2/dx^2 x^3*A(x)^2)/2! + (d^3/dx^3 x^5*A(x)^3)/3! + (d^4/dx^4 x^7*A(x)^4)/4! + (d^5/dx^5 x^9*A(x)^5)/5! + (d^6/dx^6 x^11*A(x)^6)/6! + ... + (d^n/dx^n x^(2*n-1)*A(x)^n)/n! + ...).
Related series.
Let B(x) satisfy B( x - x^2*A(x) ) = x, then
B(x) = x * exp( x*A(x) + (d/dx x^3*A(x)^2)/2! + (d^2/dx^2 x^5*B(x)^3)/3! + (d^3/dx^3 x^7*B(x)^4)/4! + (d^4/dx^4 x^9*B(x)^5)/5! + (d^5/dx^5 x^11*B(x)^6)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^n)/n! + ...)
B(x) = x + x^2 + 3*x^3 + 15*x^4 + 105*x^5 + 941*x^6 + 10227*x^7 + 130103*x^8 + 1890785*x^9 + 30848357*x^10 + ... + A360579(n)*x^n + ...
and
Series_Reversion(B(x)) = x - x^2 - x^3 - 5*x^4 - 37*x^5 - 353*x^6 - 4061*x^7 - 54221*x^8 + ... + -a(n)*x^(n+2) + ...
|
|
|
PROG
|
(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = 1 + x*sum(m=1, n, Dx(m, x^(2*m-1)*A^m/m!)) +O(x^(n+4))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
