OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..80
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 10*x^3/3! + 112*x^4/4! + 2544*x^5/5! +...
such that
A(x) = 1 + Integral B(x)^2 dx,
B(x) = 1 + Integral C(x)^4 dx,
C(x) = 1 + Integral D(x)^8 dx,
D(x) = 1 + Integral E(x)^16 dx,
E(x) = 1 + Integral F(x)^32 dx,
F(x) = 1 + Integral G(x)^64 dx, ...
The coefficients in these series begin:
A: [1, 1, 2, 10, 112, 2544, 110944, 9088160, 1395985024, ...];
B: [1, 1, 4, 44, 1048, 48472, 4171008, 663109888, 196890206720, ...];
C: [1, 1, 8, 184, 9040, 845712, 144855616, 45401856704, ...];
D: [1, 1, 16, 752, 75040, 14126752, 4830297984, 3006883867264, ...];
E: [1, 1, 32, 3040, 611392, 230931264, 157795465984, ...];
F: [1, 1, 64, 12224, 4935808, 3734695552, 5101948036608, ...];
G: [1, 1, 128, 49024, 39665920, 60075785472, 164109335366656, ...];
H: [1, 1, 256, 196352, 318046720, 963787028992, 5265107899521024, ...]; ...
To illustrate a(n) = d^n/dx^n A(x) at x=0, take successive derivatives of A=A(x):
A' = B^2;
A'' = 2*B*C^4;
A''' = 2*C^8 + 8*B*C^3*D^8;
A'''' = 24*C^7*D^8 + 24*B*C^2*D^16 + 64*B*C^3*D^7*E^16; ...
and then evaluate at x=0, where 1=A(0)=B(0)=C(0)=D(0)=E(0)=...
PROG
(PARI) {a(n)=local(A=1); for(k=0, n-1, A=1+intformal((A+x*O(x^n))^(2^(n-k)))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 24 2013
STATUS
approved