A291819 G.f. A(x) satisfies: A(x - x*A(x)) = x + 3*x*A(x).

1, 4, 24, 196, 1944, 21944, 272080, 3627412, 51288200, 761782104, 11805102064, 189901153112, 3158767322992, 54165347282960, 955189096759776, 17289056525343716, 320678326091307448, 6087009196570756488, 118109764108446889008, 2340448760238788518488, 47324471620802426563376, 975739573623235107473968, 20500725692629852174532192, 438679922664144046444438488, 9555430871381022848971028208

Offset

1,2

Links

Table of n, a(n) for n=1..25.

Formula

G.f. A(x) also satisfies:

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

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

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

Example

G.f.: A(x) = x + 4*x^2 + 24*x^3 + 196*x^4 + 1944*x^5 + 21944*x^6 + 272080*x^7 + 3627412*x^8 + 51288200*x^9 + 761782104*x^10 +...
such that  A(x - x*A(x)) = x + 3*x*A(x).
RELATED SERIES.
A(x - x*A(x)) = x + 3*x^2 + 12*x^3 + 72*x^4 + 588*x^5 + 5832*x^6 + 65832*x^7 + 816240*x^8 +...
which equals x + 3*x*A(x).
Series_Reversion( x - x*A(x) ) = x + x^2 + 6*x^3 + 49*x^4 + 486*x^5 + 5486*x^6 + 68020*x^7 + 906853*x^8 +...
which equals (1/4)*A(x) + 3*x/4.
A( (A(x) + 3*x)/4 ) = x + 5*x^2 + 38*x^3 + 369*x^4 + 4158*x^5 + 51870*x^6 + 698036*x^7 + 9974297*x^8 + 149755186*x^9 + 2345335606*x^10 +...
which equals (A(x) - x) / (A(x) + 3*x).

Prog

(PARI) {a(n) = my(A=x); for(i=1, n, A = 4*serreverse( x - x*A +x*O(x^n) ) - 3*x ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A291820, A291813, A291814, A291815, A291816, A291817, A291818.

Sequence in context: A241000 A305988 A219530 * A101370 A201338 A362355

Adjacent sequences: A291816 A291817 A291818 * A291820 A291821 A291822

Keyword

nonn

Author

Paul D. Hanna, Sep 02 2017

Status

approved