|
|
A370537
|
|
Expansion of g.f. A(x) satisfying A( x*(1 + 2*x)*A(x) )^3 = A( x*(1 + 3*x)*A(x)^2 )^2.
|
|
5
|
|
|
1, 0, 3, -10, 42, -72, 432, -1296, 3474, -11644, 48438, -119532, 385150, -1464000, 4690890, -11224776, 47891754, -153662796, 415499434, -1298706660, 5509072668, -13709346508, 41119060182, -164319443280, 531167967963, -1191393600516, 5095588415895, -18149474809934, 43672018871790
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
LINKS
|
|
|
EXAMPLE
|
G.f.: A(x) = x + 3*x^3 - 10*x^4 + 42*x^5 - 72*x^6 + 432*x^7 - 1296*x^8 + 3474*x^9 - 11644*x^10 + 48438*x^11 - 119532*x^12 + 385150*x^13 - 1464000*x^14 + 4690890*x^15 + ...
where A( x*(1 + 2*x)*A(x) )^3 = A( x*(1 + 3*x)*A(x)^2 )^2.
RELATED SERIES.
B(x) = A( x*(1 + 2*x)*A(x) )^(1/2) = A( x*(1 + 3*x)*A(x)^2 )^(1/3)
where B(x) is the g.f. of A370538, which begins
B(x) = x + x^2 + x^3 - 3*x^4 + 15*x^5 + 3*x^6 + 148*x^7 - 314*x^8 + 466*x^9 - 1980*x^10 + 13410*x^11 - 12348*x^12 + 52579*x^13 - 312347*x^14 + 898033*x^15 + ...
B(x)^2 = A( x*(1 + 2*x)*A(x) ) = x^2 + 2*x^3 + 3*x^4 - 4*x^5 + 25*x^6 + 30*x^7 + 341*x^8 - 416*x^9 + 807*x^10 - 4454*x^11 + 30125*x^12 + ...
B(x)^3 = A( x*(1 + 3*x)*A(x)^2 ) = x^3 + 3*x^4 + 6*x^5 - 2*x^6 + 33*x^7 + 75*x^8 + 607*x^9 - 189*x^10 + 1287*x^11 - 7143*x^12 + 48735*x^13 + ...
B(x)^6 = A( x*(1 + 2*x)*A(x) )^3 = x^6 + 6*x^7 + 21*x^8 + 32*x^9 + 90*x^10 + 324*x^11 + 2064*x^12 + 4032*x^13 + 9513*x^14 - 6310*x^15 + 116499*x^16 + ...
|
|
PROG
|
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); Ax = x*Ser(A);
A[#A] = polcoeff( subst(Ax, x, x*(1 + 2*x)*Ax )^3 - subst(Ax, x, x*(1 + 3*x)*Ax^2 )^2, #A+5); ); A[n]}
for(n=1, 30, print1(a(n), ", "))
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|