%I #14 Sep 30 2019 09:31:25
%S 1,4,52,1228,42652,1972324,114581476,8051020348,666126945340,
%T 63620722928308,6907454641512244,842227742687112604,
%U 114192665828161184332,17076069626235659815108,2796969496541969481342100,498871283058754285439126092,96403472225110465517090352700,20094942949266343527252229063204,4500802213556155723422379457382916,1079478060677848794106956676648220860
%N E.g.f. satisfies: A(x - Integral 3*A(x) dx) = x + Integral A(x) dx.
%F E.g.f. A(x) satisfies:
%F (1) A(x - Integral 3*A(x) dx) = x + Integral A(x) dx.
%F (2) A(x) = x + 4 * G( (3*A(x) + x)/4 ), where G(x) = Integral 3*A(x) dx.
%F (3) A(x) = -x/3 + 4/3 * Series_Reversion(x - Integral 3*A(x) dx).
%F (4) A( (3*A(x) + x)/4 ) = (A'(x) - 1)/(3*A'(x) + 1).
%F (5) A'(x - Integral 3*A(x) dx) = (1 + A(x))/(1 - 3*A(x)).
%F a(n) = Sum_{k=0..n-1} A277410(n,k) * 3^k * 4^(n-k-1).
%e E.g.f.: A(x) = x + 4*x^2/2! + 52*x^3/3! + 1228*x^4/4! + 42652*x^5/5! + 1972324*x^6/6! + 114581476*x^7/7! + 8051020348*x^8/8! + 666126945340*x^9/9! + 63620722928308*x^10/10! + 6907454641512244*x^11/11! + 842227742687112604*x^12/12! +...
%e Let G(x) = Integral A(x) dx, then A(x - 3*G(x)) = x + G(x) where
%e G(x) = x^2/2! + 4*x^3/3! + 52*x^4/4! + 1228*x^5/5! + 42652*x^6/6! + 1972324*x^7/7! + 114581476*x^8/8! + 8051020348*x^9/9! + 666126945340*x^10/10! + 63620722928308*x^11/11! + 6907454641512244*x^12/12! +...
%e Also, A(x) = x + 4 * G( (3*A(x) + x)/4 ).
%e RELATED SERIES.
%e We have (3*A(x) + x)/4 = Series_Reversion( x - Integral 3*A(x) dx ), where
%e (3*A(x) + x)/4 = x + 3*x^2/2! + 39*x^3/3! + 921*x^4/4! + 31989*x^5/5! + 1479243*x^6/6! + 85936107*x^7/7! + 6038265261*x^8/8! + 499595209005*x^9/9! + 47715542196231*x^10/10! + 5180590981134183*x^11/11! + 631670807015334453*x^12/12! +...
%e Further, A( (3*A(x) + x)/4 ) = (A'(x) - 1)/(3*A'(x) + 1), which begins
%e A( (3*A(x) + x)/4 ) = x + 7*x^2/2! + 127*x^3/3! + 3817*x^4/4! + 161881*x^5/5! + 8924923*x^6/6! + 608517595*x^7/7! + 49615007497*x^8/8! + 4722073055173*x^9/9! + 515139762620935*x^10/10! + 63506672456719651*x^11/11! + 8747178021763399909*x^12/12! +...
%t m = 21; A[_] = 0;
%t Do[A[x_] = -x/3 + 4/3 InverseSeries[x-Integrate[3 A[x], x] + O[x]^m], {m}];
%t CoefficientList[A[x], x]*Range[0, m-1]! // Rest (* _Jean-François Alcover_, Sep 30 2019 *)
%o (PARI) /* A(x) = x + (p+q)*G((p*A(x) + q*x)/(p+q)) ; G(x) = Integral A(x) dx: */
%o {a(n, p=3, q=1) = my(A=x, G); for(i=1, n, G = intformal(A +x*O(x^n)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q)) +x*O(x^n)); n!*polcoeff(A, n)}
%o for(n=1, 30, print1(a(n, 3, 1), ", "))
%o (PARI) /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
%o {a(n, p=3, q=1) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); G=intformal(F); A[#A] = -polcoeff(subst(F, x, x - p*G) - q*G, #A) ); n!*A[n]}
%o for(n=1, 30, print1(a(n, 3, 1), ", "))
%o (PARI) /* Informal code to generate the first N terms: */
%o {N=20; p=3; q=1; A=x; for(i=1, N, G=intformal(A +x*O(x^N)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q))); Vec(serlaplace(A))}
%Y Cf. A277410, A210949, A277403, A279843, A279844, A279845, A280570, A280572, A280573, A280574, A280575.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jan 05 2017