|
| |
|
|
A275765
|
|
G.f. satisfies: A(x - A(x)^2) = x + A(x)^2.
|
|
25
|
|
|
|
1, 2, 12, 106, 1148, 14156, 191400, 2775930, 42585412, 684496988, 11449962008, 198331811356, 3543990791480, 65136985937096, 1228531761076208, 23733123786608826, 468887742020767788, 9461919438245032500, 194817077269127033944, 4089069139317823277548, 87426000975842460304792, 1902787414323673070857528, 42133267254272433484761584, 948695717599714654940068604, 21712101305047777916075831096, 504865916349551192319293625016
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. A(x) also satisfies:
(1) A(x) = x + 2 * A( x/2 + A(x)/2 )^2.
(2) A(x) = -x + 2 * Series_Reversion(x - A(x)^2).
(3) R(x) = -x + 2 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/2 - R(x)/2 ) ) = x/2 + R(x)/2, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*2^(n-k).
|
|
|
EXAMPLE
|
G.f.: A(x) = x + 2*x^2 + 12*x^3 + 106*x^4 + 1148*x^5 + 14156*x^6 + 191400*x^7 + 2775930*x^8 + 42585412*x^9 + 684496988*x^10 + 11449962008*x^11 + 198331811356*x^12 +...
such that A(x - A(x)^2) = x + A(x)^2.
RELATED SERIES.
Series_Reversion(x - A(x)^2) = x + x^2 + 6*x^3 + 53*x^4 + 574*x^5 + 7078*x^6 + 95700*x^7 + 1387965*x^8 + 21292706*x^9 + 342248494*x^10 +...
which equals (A(x) + x)/2.
A( (A(x) + x)/2 ) = x + 3*x^2 + 22*x^3 + 221*x^4 + 2634*x^5 + 35086*x^6 + 506356*x^7 + 7773279*x^8 + 125441594*x^9 + 2110832382*x^10 +...
which equals sqrt( (A(x) - x)/2 ).
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 2*x^2 - 4*x^3 - 26*x^4 - 228*x^5 - 2396*x^6 - 28440*x^7 - 369114*x^8 - 5135468*x^9 - 75602108*x^10 - 1167066216*x^11 - 18768202924*x^12 +...
then Series_Reversion(x + A(x)^2) = x/2 + R(x)/2.
|
|
|
MATHEMATICA
|
m = 26; A[_] = 0;
Do[A[x_] = x + 2 A[x/2 + A[x]/2]^2 + O[x]^(m+1) // Normal, {m+1}];
|
|
|
PROG
|
(PARI) {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-F^2) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
