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%I #12 Sep 30 2019 09:31:36
%S 1,5,40,525,10025,253475,8015725,305359050,13645726250,701304298375,
%T 40822454374125,2658840618527250,191861336190647375,
%U 15213199343853357500,1316408013706224687500,123576861126283832953125,12521371849855149886590625,1363361618975383978443843750,158900334287408210286438971875,19755940413686794723417400000000,2612146114817877629253999384562500,366294181903982533559997504649828125
%N E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral 4*A(x) dx.
%F E.g.f. A(x) satisfies:
%F (1) A(x - Integral A(x) dx) = x + Integral 4*A(x) dx.
%F (2) A(x) = x + 5 * G( (A(x) + 4*x)/5 ), where G(x) = Integral A(x) dx.
%F (3) A(x) = -4*x + 5 * Series_Reversion(x - Integral A(x) dx).
%F (4) A( (A(x) + 4*x)/5 ) = (A'(x) - 1)/(A'(x) + 4).
%F (5) A'(x - Integral A(x) dx) = (1 + 4*A(x))/(1 - A(x)).
%F a(n) = Sum_{k=0..n-1} A277410(n,k) * 5^(n-k-1).
%e E.g.f.: A(x) = x + 5*x^2/2! + 40*x^3/3! + 525*x^4/4! + 10025*x^5/5! + 253475*x^6/6! + 8015725*x^7/7! + 305359050*x^8/8! + 13645726250*x^9/9! + 701304298375*x^10/10! + 40822454374125*x^11/11! + 2658840618527250*x^12/12! +...
%e Let G(x) = Integral A(x) dx, then A(x - G(x)) = x + 4*G(x) where
%e G(x) = x^2/2! + 5*x^3/3! + 40*x^4/4! + 525*x^5/5! + 10025*x^6/6! + 253475*x^7/7! + 8015725*x^8/8! + 305359050*x^9/9! + 13645726250*x^10/10! + 701304298375*x^11/11! + 40822454374125*x^12/12! +...
%e Also, A(x) = x + 5 * G( (A(x) + 4*x)/5 ).
%e RELATED SERIES.
%e We have (A(x) + 4*x)/5 = Series_Reversion( x - Integral A(x) dx ), where
%e (A(x) + 4*x)/5 = x + x^2/2! + 8*x^3/3! + 105*x^4/4! + 2005*x^5/5! + 50695*x^6/6! + 1603145*x^7/7! + 61071810*x^8/8! + 2729145250*x^9/9! + 140260859675*x^10/10! + 8164490874825*x^11/11! + 531768123705450*x^12/12! +...
%e Further, A( (A(x) + 4*x)/5 ) = (A'(x) - 1)/(A'(x) + 4), which begins
%e A( (A(x) + 4*x)/5 ) = x + 6*x^2/2! + 63*x^3/3! + 1045*x^4/4! + 24105*x^5/5! + 716195*x^6/6! + 26137820*x^7/7! + 1134457060*x^8/8! + 57203895725*x^9/9! + 3292221321425*x^10/10! + 213282348138700*x^11/11! + 15380885339509825*x^12/12! +...
%t m = 23; A[_] = 0;
%t Do[A[x_] = -4 x + 5 InverseSeries[x - Integrate[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=1, q=4) = 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, 1, 4), ", "))
%o (PARI) /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
%o {a(n, p=1, q=4) = 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, 1, 4), ", "))
%o (PARI) /* Informal code to generate the first N terms: */
%o {N=20; p=1; q=4; 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, A280571, A280573, A280574, A280575.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jan 05 2017
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