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A291819 G.f. A(x) satisfies: A(x - x*A(x)) = x + 3*x*A(x). 7

%I #5 Sep 15 2017 22:09:52

%S 1,4,24,196,1944,21944,272080,3627412,51288200,761782104,11805102064,

%T 189901153112,3158767322992,54165347282960,955189096759776,

%U 17289056525343716,320678326091307448,6087009196570756488,118109764108446889008,2340448760238788518488,47324471620802426563376,975739573623235107473968,20500725692629852174532192,438679922664144046444438488,9555430871381022848971028208

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

%F G.f. A(x) also satisfies:

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

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

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

%e 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 +...

%e such that A(x - x*A(x)) = x + 3*x*A(x).

%e RELATED SERIES.

%e 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 +...

%e which equals x + 3*x*A(x).

%e 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 +...

%e which equals (1/4)*A(x) + 3*x/4.

%e 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 +...

%e which equals (A(x) - x) / (A(x) + 3*x).

%o (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)}

%o for(n=1, 30, print1(a(n), ", "))

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

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 02 2017

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