A291817 G.f. A(x) satisfies: A(x - 3*x*A(x)) = x + x*A(x).

1, 4, 40, 580, 10312, 209752, 4707952, 114128308, 2946787192, 80268150808, 2290811949904, 68149759850680, 2105074730357968, 67308231895605520, 2222306773263886624, 75615701295449074084, 2647156154616207962920, 95215874465318554556776, 3514739264129342710455184, 133012394550946993742673304, 5156112210399927390763631056, 204570581814658024128260509360

Offset

1,2

Links

Table of n, a(n) for n=1..22.

Formula

G.f. A(x) also satisfies:

(1) A(x) = (4/3)*Series_Reversion( x - 3*x*A(x) ) - x/3.

(2) A( (3*A(x) + x)/4 ) = (A(x) - x) / (3*A(x) + x).

a(n) = Sum_{k=0..n-1} A291820(n, k) * 3^k * 4^(n-k-1).

Example

G.f.: A(x) = x + 4*x^2 + 40*x^3 + 580*x^4 + 10312*x^5 + 209752*x^6 + 4707952*x^7 + 114128308*x^8 + 2946787192*x^9 + 80268150808*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 + 4*x^3 + 40*x^4 + 580*x^5 + 10312*x^6 + 209752*x^7 + 4707952*x^8 +...
which equals x + x*A(x).
Series_Reversion( x - 3*x*A(x) ) = x + 3*x^2 + 30*x^3 + 435*x^4 + 7734*x^5 + 157314*x^6 + 3530964*x^7 + 85596231*x^8 +...
which equals (3/4)*A(x) + x/4.
A( (3*A(x) + x)/4 )  = x + 7*x^2 + 94*x^3 + 1651*x^4 + 33886*x^5 + 773458*x^6 + 19117780*x^7 + 503529979*x^8 + 13983485770*x^9 + 406470316978*x^10 +...
which equals (A(x) - x) / (3*A(x) + x).

Prog

(PARI) {a(n) = my(A=x); for(i=1, n, A = (4/3)*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, A291816, A291818, A291819.

Sequence in context: A276362 A302178 A272261 * A128573 A052675 A141010

Adjacent sequences: A291814 A291815 A291816 * A291818 A291819 A291820

Keyword

nonn

Author

Paul D. Hanna, Sep 02 2017

Status

approved