|
| |
|
|
A274737
|
|
G.f. A(x) satisfies: A(x)^2 - 6*A(x)^3 = A(x^2).
|
|
2
|
|
|
|
1, 3, 24, 225, 2451, 28584, 350811, 4456971, 58132194, 773773785, 10468458657, 143528736888, 1989864432072, 27848242441521, 392899157668962, 5582238744258009, 79799757710924847, 1146961633262521734, 16564959319032187542, 240274065224349972819, 3498743054027725572015, 51126484223473738706979, 749497753666229701655097, 11019579243648283996016040, 162451851753106862734656771, 2400803315482934131290808344
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. A(x) satisfies: A(B(x)^2) = x^2 - 6*x^3, where A(B(x)) = x.
|
|
|
EXAMPLE
|
G.f.: A(x) = x + 3*x^2 + 24*x^3 + 225*x^4 + 2451*x^5 + 28584*x^6 + 350811*x^7 + 4456971*x^8 + 58132194*x^9 + 773773785*x^10 +...
such that A(x)^2 - 6*A(x)^3 = A(x^2).
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 57*x^4 + 594*x^5 + 6828*x^6 + 82674*x^7 + 1041399*x^8 + 13493790*x^9 + 178715343*x^10 + 2408259060*x^11 + 32912262864*x^12 +...
A(x)^3 = x^3 + 9*x^4 + 99*x^5 + 1134*x^6 + 13779*x^7 + 173529*x^8 + 2248965*x^9 + 29785482*x^10 + 401376510*x^11 + 5485372380*x^12 +...
Let B(x) denote the series reversion of g.f. A(x), so that A(B(x)) = x, where
B(x) = x - 3*x^2 - 6*x^3 - 75*x^5 - 171*x^6 - 1287*x^7 - 4239*x^8 - 23289*x^9 - 107001*x^10 - 585468*x^11 - 2852334*x^12 - 15659352*x^13 - 80867160*x^14 +...
Note that g.f. A(x) and B(x) satisfy:
(1) A(B(x)^2) = C(x) = x^2 - 6*x^3.
(2) A(B(x)^4) = C(C(x)) = x^4 - 12*x^5 + 30*x^6 + 108*x^7 - 648*x^8 + 1296*x^9.
(3) A(B(x)^8) = C(C(C(x))).
|
|
|
PROG
|
(PARI) /* From A(B(x)^2) = x^2 - 6*x^3, where A(B(x)) = x: */
{a(n) = my(A=[1, 3], F, B); for(i=1, n, A=concat(A, 0); F=x*Ser(A); B=serreverse(F); A[#A] = Vec(subst(F, x, B^2))[#A]/2); A[n]}
for(n=1, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
