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%I #7 Sep 02 2017 04:16:05
%S 1,1,9,109,1569,25481,454105,8730373,178996865,3881556561,88477557289,
%T 2109927671453,52443846331297,1354646602217945,36275862587452281,
%U 1005099719255707829,28765965099599741953,849204340574458575777,25827102287376124267593,808349897942417046805197,26011340193853765710238241,859773626049480606121078057,29168437337569276216572259097
%N G.f. A(x) satisfies: A(x - 4*x*A(x)) = x - 3*x*A(x).
%F G.f. A(x) also satisfies:
%F (1) A(x) = (1/4)*Series_Reversion( x - 4*x*A(x) ) + 3*x/4.
%F (2) A( 4*A(x) - 3*x) = (A(x) - x) / (4*A(x) - 3*x).
%F a(n) = Sum_{k=0..n-1} A291820(n, k) * 4^k.
%e 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 +...
%e such that A(x - 4*x*A(x)) = x - 3*x*A(x).
%e RELATED SERIES.
%e 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 +...
%e which equals x - 3*x*A(x).
%e 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 +...
%e which equals 4*A(x) - 3*x.
%e 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 +...
%e which equals (A(x) - x) / (4*A(x) - 3*x).
%o (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)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A291820, A291813, A291814, A291816, A291817, A291818, A291819, A276358, A291743, A291744.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Sep 02 2017
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