A291815 G.f. A(x) satisfies: A(x - 4*x*A(x)) = x - 3*x*A(x).

1, 1, 9, 109, 1569, 25481, 454105, 8730373, 178996865, 3881556561, 88477557289, 2109927671453, 52443846331297, 1354646602217945, 36275862587452281, 1005099719255707829, 28765965099599741953, 849204340574458575777, 25827102287376124267593, 808349897942417046805197, 26011340193853765710238241, 859773626049480606121078057, 29168437337569276216572259097

Offset

1,3

Links

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

Formula

G.f. A(x) also satisfies:

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

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

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

Example

G.f.: A(x) = x + x^2 + 9*x^3 + 109*x^4 + 1569*x^5 + 25481*x^6 + 454105*x^7 + 8730373*x^8 + 178996865*x^9 + 3881556561*x^10 + 88477557289*x^11 + 2109927671453*x^12 +...
such that  A(x - 4*x*A(x)) = x - 3*x*A(x).
RELATED SERIES.
A(x - 4*x*A(x)) = x - 3*x^2 - 3*x^3 - 27*x^4 - 327*x^5 - 4707*x^6 - 76443*x^7 +...
which equals x - 3*x*A(x).
Series_Reversion( x - 4*x*A(x) ) = x + 4*x^2 + 36*x^3 + 436*x^4 + 6276*x^5 + 101924*x^6 + 1816420*x^7 + 34921492*x^8 +...
which equals 4*A(x) - 3*x.
A( 4*A(x) - 3*x )  = x + 5*x^2 + 53*x^3 + 741*x^4 + 12153*x^5 + 222405*x^6 + 4421501*x^7 + 93949493*x^8 + 2110952881*x^9 + 49786323589*x^10 + 1225967873349*x^11 + 31395927333829*x^12 +...
which equals (A(x) - x) / (4*A(x) - 3*x).

Prog

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

Crossrefs

Cf. A291820, A291813, A291814, A291816, A291817, A291818, A291819, A276358, A291743, A291744.

Sequence in context: A288692 A358601 A349335 * A261502 A359731 A105974

Adjacent sequences: A291812 A291813 A291814 * A291816 A291817 A291818

Keyword

nonn

Author

Paul D. Hanna, Sep 02 2017

Status

approved