A273203 G.f. A(x) satisfies: A(x^2*B(x)) = x^3 - x^4, where A(B(x)) = x.

1, 1, 2, 6, 18, 57, 192, 666, 2362, 8548, 31422, 116967, 440100, 1671097, 6395116, 24640671, 95511110, 372179427, 1457127788, 5728994316, 22610712406, 89546822784, 355755807944, 1417433615730, 5662390311408, 22675286872185, 91008415721478, 366028366965891, 1474988243789244, 5954505719829317, 24078786607483276, 97523329765129081, 395571627038520046, 1606741060249697567

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..515

Formula

G.f. A(x) satisfies: A(x*A(x)^2) = A(x)^3 - A(x)^4.

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 18*x^5 + 57*x^6 + 192*x^7 + 666*x^8 + 2362*x^9 + 8548*x^10 + 31422*x^11 + 116967*x^12 +...
such that A(x^2*B(x)) = x^3 - x^4, where A(B(x)) = x.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 52*x^6 + 174*x^7 + 606*x^8 + 2160*x^9 + 7832*x^10 + 28840*x^11 + 107541*x^12 + 405178*x^13 +...
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 108*x^7 + 381*x^8 + 1376*x^9 + 5049*x^10 + 18750*x^11 + 70398*x^12 + 266799*x^13 + 1019196*x^14 +...
A(x)^4 = x^4 + 4*x^5 + 14*x^6 + 52*x^7 + 193*x^8 + 716*x^9 + 2684*x^10 + 10148*x^11 + 38636*x^12 + 148096*x^13 + 571182*x^14 + 2215072*x^15 +...
A(x*A(x)^2) = x^3 + 2*x^4 + 5*x^5 + 17*x^6 + 56*x^7 + 188*x^8 + 660*x^9 + 2365*x^10 + 8602*x^11 + 31762*x^12 + 118703*x^13 + 448014*x^14 + 1705514*x^15 + 6541232*x^16 + 25251188*x^17 + 98036913*x^18 + 382565722*x^19 + 1499669634*x^20 +...
where A(x*A(x)^2) = A(x)^3 - A(x)^4.
Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then
B(x) = x - x^2 - x^4 + 2*x^5 - x^6 - x^10 + 4*x^11 - 6*x^12 + 6*x^13 - 11*x^14 + 20*x^15 - 21*x^16 + 16*x^17 + ... + A350433(n)*x^n +...
such that A(x^2*B(x)) = x^3 - x^4,
also, B(x) = B(x^3 - x^4)/x^2.
From Paul D. Hanna, Dec 31 2021: (Start)
GENERATING METHOD.
Define F(n), a polynomial in x of order 4^(n-1), by the following recurrence:
F(0) = x,
F(1) = (1 - x),
F(2) = (1 - x^3 * (1-x)),
F(3) = (1 - x^9 * (1-x)^3 * F(2)),
F(4) = (1 - x^27 * (1-x)^9 * F(2)^3 * F(3)),
F(5) = (1 - x^81 * (1-x)^27 * F(2)^9 * F(3)^3 * F(4)),
...
F(n+1) = 1 - (1 - F(n))^3 * F(n)
...
Then the series reversion B(x) of g.f. A(x) equals the infinite product:
B(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
that is,
B(x) = x * (1-x) * (1 - x^3*(1-x)) * (1 - x^9*(1-x)^3*(1 - x^3*(1-x))) * (1 - x^27*(1-x)^9*(1 - x^3*(1-x))^3*(1 - x^9*(1-x)^3*(1 - x^3*(1-x)))) * ...
(End)

Prog

(PARI) {a(n) = my(A=[1, 1], F, B); for(i=1, n, A=concat(A, 0); F=x*Ser(A); B=serreverse(F); A[#A] = Vec(subst(F, x, x^2*B))[#A]); A[n]}
for(n=1, 50, print1(a(n), ", "))

Crossrefs

Cf. A350433 (inverse), A273162.

Sequence in context: A339044 A125305 A369621 * A148458 A148459 A324166

Adjacent sequences: A273200 A273201 A273202 * A273204 A273205 A273206

Keyword

nonn

Author

Paul D. Hanna, May 17 2016

Status

approved