login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A247224
G.f. A(x) satisfies: A(x) = B(x)*(A(x) - x*C(x)) where B(x) = A(x/B(x)) and C(x) = A(x*C(x)).
1
1, 1, 2, 6, 26, 149, 1024, 7965, 68192, 632724, 6294190, 66579501, 744194484, 8747497833, 107718981328, 1385436413289, 18563761993762, 258579817821799, 3737335096804136, 55957031888334621, 866632465992896412, 13865193902724224273, 228875892203793317404, 3893773927147402337094
OFFSET
0,3
COMMENTS
Compare to: G(x) = Series_Reversion( x - Series_Reversion(x*G(x)) * x*G(x) )/x, which is satisfied by the g.f. G(x) = 1 + x*G(x) * G(x*G(x)) of A030266 with offset 0.
LINKS
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Series_Reversion( x - Series_Reversion(x/A(x)) * x/A(x) )/x.
(2) A(x) = x/Series_Reversion( (x - Series_Reversion(x*A(x))) * A(x)/x ).
Given B(x) = A(x/B(x)) and C(x) = A(x*C(x)), then:
(3.a) A(x) = B(x*A(x)) and A(x) = C(x/A(x)),
(3.b) B(x) = x/Series_Reversion(x*A(x)),
(3.c) C(x) = Series_Reversion(x/A(x))/x,
(3.d) B(x) = A(x)/(A(x) - x*C(x)),
(3.e) C(x) = A(x)*(1 - 1/B(x))/x.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 26*x^4 + 149*x^5 + 1024*x^6 +...
Let B(x) = A(x/B(x)) and C(x) = A(x*C(x)), where B(x) and C(x) begin:
B(x) = 1 + x + x^2 + 2*x^3 + 9*x^4 + 53*x^5 + 354*x^6 + 2651*x^7 + 21951*x^8 + 197666*x^9 + 1911091*x^10 + 19665622*x^11 + 214060860*x^12 +...
C(x) = 1 + x + 3*x^2 + 13*x^3 + 71*x^4 + 460*x^5 + 3399*x^6 + 27867*x^7 + 248789*x^8 + 2388199*x^9 + 24432778*x^10 + 264682253*x^11 + 3021086129*x^12 +...
then A(x) = B(x) * (A(x) - x*C(x)).
PROG
(PARI) {a(n)=local(A=1); for(i=1, n, A=serreverse(x-x/A*serreverse(x/(A +x^2*O(x^n))))/x); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Sequence in context: A125224 A052844 A375629 * A052859 A103937 A159311
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 19 2014
STATUS
approved