A249791 Given g.f. A(x), let B(x) = 1 + x*A(x)^2 and C(x) = 1 + x*A(x)^3, then B(x*C(x)) = C(x) and C(x/B(x)) = B(x).

1, 1, 5, 44, 530, 7911, 139129, 2798844, 63178500, 1578855377, 43245568061, 1288116498182, 41457303331745, 1433966498431138, 53058288363011906, 2091593330699875406, 87527301512425974261, 3875685191976323542974, 181061755084572933223563, 8900849566241379829936126

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) satisfies:

(1) A( x*(1 + x*A(x)^3) )^2 * (1 + x*A(x)^3) = A(x)^3.

(2) A( x/(1 + x*A(x)^2) )^3 / (1 + x*A(x)^2) = A(x)^2.

(3) x / Series_Reversion( x*(1 + x*A(x)^3) ) = 1 + x*A(x)^2.

(4) (1/x) * Series_Reversion( x/(1 + x*A(x)^2) ) = 1 + x*A(x)^3.

Example

G.f.: A(x) = 1 + x + 5*x^2 + 44*x^3 + 530*x^4 + 7911*x^5 + 139129*x^6 +...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 11*x^2 + 98*x^3 + 1173*x^4 + 17322*x^5 + 301316*x^6 +...
A(x)^3 = 1 + 3*x + 18*x^2 + 163*x^3 + 1944*x^4 + 28440*x^5 + 489596*x^6 +...
A(x)^4 = 1 + 4*x + 26*x^2 + 240*x^3 + 2859*x^4 + 41492*x^5 + 707330*x^6 +...
A(x)^2 + x*A(x)^4 = 1 + 3*x + 15*x^2 + 124*x^3 + 1413*x^4 + 20181*x^5 +...
A(x)^3/(1 + x*A(x)^3) = 1 + 2*x + 13*x^2 + 126*x^3 + 1580*x^4 + 23978*x^5 +...
A( x*(1 + x*A(x)^3) )^2 = 1 + 2*x + 13*x^2 + 126*x^3 + 1580*x^4 + 23978*x^5 +...
A( x/(1 + x*A(x)^2) )^3 = 1 + 3*x + 15*x^2 + 124*x^3 + 1413*x^4 + 20181*x^5 +...
A( x*(1 + x*A(x)^3) ) = 1 + x + 6*x^2 + 57*x^3 + 715*x^4 + 10932*x^5 +...
A( x/(1 + x*A(x)^2) ) = 1 + x + 4*x^2 + 33*x^3 + 385*x^4 + 5644*x^5 +...
The table of coefficients in (1 + x*A(x)^2)^n begins:
n=1: [1, 1,  2,  11,   98,  1173,  17322,  301316,  6001696, ...];
n=2: [1, 2,  5,  26,  222,  2586,  37503,  644124, 12710722, ...];
n=3: [1, 3,  9,  46,  378,  4284,  60977, 1033614, 20201490, ...];
n=4: [1, 4, 14,  72,  573,  6320,  88246, 1475664, 28556726, ...];
n=5: [1, 5, 20, 105,  815,  8756, 119890, 1976935, 37868480, ...];
n=6: [1, 6, 27, 146, 1113, 11664, 156578, 2544978, 48239262, ...];
n=7: [1, 7, 35, 196, 1477, 15127, 199080, 3188354, 59783318, ...];
n=8: [1, 8, 44, 256, 1918, 19240, 248280, 3916768, 72628061, ...];
n=9: [1, 9, 54, 327, 2448, 24111, 305190, 4741218, 86915673, ...]; ...
in which the main diagonal generates coefficients in (1 + x*A(x)^3):
[1, 2/2, 9/3, 72/4, 815/5, 11664/6, 199080/7, 3916768/8, 86915673/9, ...]
= [1, 1, 3, 18, 163, 1944, 28440, 489596, 9657297, ...].

Prog

(PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0);
A[#A]=Vec(serreverse(x/(1+x*Ser(A)^2))/x - x*Ser(A)^3)[#A+1]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0);
A[#A]=-Vec(x/serreverse(x*(1+x*Ser(A)^3)) - x*Ser(A)^2)[#A+1]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A252830 A301434 A232192 * A215648 A195242 A243697

Adjacent sequences: A249788 A249789 A249790 * A249792 A249793 A249794

Keyword

nonn

Author

Paul D. Hanna, Nov 11 2014

Status

approved