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A291814 G.f. A(x) satisfies: A(x - 3*x*A(x)) = x - 2*x*A(x). 7
1, 1, 7, 67, 769, 10009, 143359, 2218255, 36625657, 639659737, 11741022235, 225390779647, 4508109360985, 93665093491381, 2016669357747667, 44905700922069463, 1032419000661778213, 24472819932819733957, 597384952530618840715, 15000294032677574361955, 387082666821619977435277, 10256260095368150955828565, 278811213889895147327704519, 7770474960716476086765483619 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
FORMULA
G.f. A(x) also satisfies:
(1) A(x) = (1/3)*Series_Reversion( x - 3*x*A(x) ) + 2*x/3.
(2) A( 3*A(x) - 2*x) ) = (A(x) - x) / (3*A(x) - 2*x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 3^k.
EXAMPLE
G.f.: A(x) = x + x^2 + 7*x^3 + 67*x^4 + 769*x^5 + 10009*x^6 + 143359*x^7 + 2218255*x^8 + 36625657*x^9 + 639659737*x^10 + 11741022235*x^11 + 225390779647*x^12 +...
such that A(x - 3*x*A(x)) = x - 2*x*A(x).
RELATED SERIES.
A(x - 3*x*A(x)) = x - 2*x^2 - 2*x^3 - 14*x^4 - 134*x^5 - 1538*x^6 - 20018*x^7 +...
which equals x - 2*x*A(x).
Series_Reversion( x - 3*x*A(x) ) = x + 3*x^2 + 21*x^3 + 201*x^4 + 2307*x^5 + 30027*x^6 + 430077*x^7 + 6654765*x^8 +...
which equals 3*A(x) - 2*x.
A( 3*A(x) - 2*x ) = x + 4*x^2 + 34*x^3 + 382*x^4 + 5038*x^5 + 74134*x^6 + 1184650*x^7 + 20224990*x^8 + 364994554*x^9 + 6911857450*x^10 + 136622440786*x^11 + 2807805653098*x^12 +...
which equals (A(x) - x) / (3*A(x) - 2*x).
PROG
(PARI) {a(n) = my(A=x); for(i=1, n, A = (1/3)*serreverse( x - 3*x*A +x*O(x^n) ) + 2*x/3 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
Sequence in context: A371398 A364924 A082578 * A253386 A082698 A182127
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 02 2017
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)