A291816 G.f. A(x) satisfies: A(x - 3*x*A(x)) = x - x*A(x).

1, 2, 16, 182, 2524, 39992, 699520, 13231034, 266985280, 5694001172, 127481465536, 2981125793144, 72532301230672, 1830526849868000, 47802726801684544, 1289123410465365782, 35841130838977837348, 1025903099063974343984, 30195807234087904770952, 912951678159786641659796, 28327442752528049524839856, 901289532361030971832330544, 29382621186595702051011638128

Offset

1,2

Links

Table of n, a(n) for n=1..23.

Formula

G.f. A(x) also satisfies:

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

(2) A( (3*A(x) - x)/2 ) = (A(x) - x) / (3*A(x) - x).

a(n) = Sum_{k=0..n-1} A291820(n, k) * 3^k * 2^(n-k-1).

Example

G.f.: A(x) = x + 2*x^2 + 16*x^3 + 182*x^4 + 2524*x^5 + 39992*x^6 + 699520*x^7 + 13231034*x^8 + 266985280*x^9 + 5694001172*x^10 +...
such that  A(x - 3*x*A(x)) = x - x*A(x).
RELATED SERIES.
A(x - 3*x*A(x)) = x - x^2 - 2*x^3 - 16*x^4 - 182*x^5 - 2524*x^6 - 39992*x^7 - 699520*x^8 +...
which equals x - x*A(x).
Series_Reversion( x - 3*x*A(x) ) = x + 3*x^2 + 24*x^3 + 273*x^4 + 3786*x^5 + 59988*x^6 + 1049280*x^7 + 19846551*x^8 +...
which equals (3/2)*A(x) - x/2.
A( (3*A(x) - x)/2 )  = x + 5*x^2 + 52*x^3 + 713*x^4 + 11458*x^5 + 205160*x^6 + 3984304*x^7 + 82576109*x^8 + 1807215616*x^9 + 41461917398*x^10 +...
which equals (A(x) - x) / (3*A(x) - x).

Prog

(PARI) {a(n) = my(A=x); for(i=1, n, A = (2/3)*serreverse( x - 3*x*A +x*O(x^n) ) + x/3 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A291820, A291813, A291814, A291815, A291817, A291818, A291819, A276358, A291743, A291744.

Sequence in context: A363311 A052606 A011553 * A123898 A118644 A183205

Adjacent sequences: A291813 A291814 A291815 * A291817 A291818 A291819

Keyword

nonn

Author

Paul D. Hanna, Sep 02 2017

Status

approved