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A279844 E.g.f. satisfies: A(x - Integral 2*A(x) dx) = x + Integral A(x) dx.

%I A279844 #15 Oct 20 2019 09:38:10
%S A279844 1,3,27,441,10593,338715,13603923,660689217,37773985257,2492351980659,
%T A279844 186888829248171,15733456044557193,1472423968474987185,
%U A279844 151932311464679521803,17166519680224611739203,2111435499783771418877073,281279117575497421255121721,40406056752677361995435879907,6234806360224720540046684747547,1029860015641146082486445487150681
%N A279844 E.g.f. satisfies: A(x - Integral 2*A(x) dx) = x + Integral A(x) dx.
%H A279844 Paul D. Hanna, <a href="/A279844/b279844.txt">Table of n, a(n) for n = 1..150</a>
%F A279844 E.g.f. A(x) satisfies:
%F A279844 (1) A(x - Integral 2*A(x) dx) = x + Integral A(x) dx.
%F A279844 (2) A(x) = x + 3 * G( (2*A(x) + x)/3 ), where G(x) = Integral A(x) dx.
%F A279844 (3) A(x) = -x/2 + 3/2 * Series_Reversion(x - Integral 2*A(x) dx).
%F A279844 (4) A( (2*A(x) + x)/3 ) = (A'(x) - 1)/(2*A'(x) + 1).
%F A279844 (5) A'(x - Integral 2*A(x) dx) = (1 + A(x))/(1 - 2*A(x)).
%F A279844 a(n) = Sum_{k=0..n-1} A277410(n,k) * 2^k * 3^(n-k-1).
%e A279844 E.g.f.: A(x) = x + 3*x^2/2! + 27*x^3/3! + 441*x^4/4! + 10593*x^5/5! + 338715*x^6/6! + 13603923*x^7/7! + 660689217*x^8/8! + 37773985257*x^9/9! + 2492351980659*x^10/10! + 186888829248171*x^11/11! + 15733456044557193*x^12/12! +...
%e A279844 Let G(x) = Integral A(x) dx, then A(x - 2*G(x)) = x + G(x) where
%e A279844 G(x) = x^2/2! + 3*x^3/3! + 27*x^4/4! + 441*x^5/5! + 10593*x^6/6! + 338715*x^7/7! + 13603923*x^8/8! + 660689217*x^9/9! + 37773985257*x^10/10! +...
%e A279844 Also, A(x) = x + 3 * G( (2*A(x) + x)/3 ).
%e A279844 RELATED SERIES.
%e A279844 We have (2*A(x) + x)/3 = Series_Reversion( x - Integral 2*A(x) dx ), where
%e A279844 (2*A(x) + x)/3 = x + 2*x^2/2! + 18*x^3/3! + 294*x^4/4! + 7062*x^5/5! + 225810*x^6/6! + 9069282*x^7/7! + 440459478*x^8/8! + 25182656838*x^9/9! + 1661567987106*x^10/10! + 124592552832114*x^11/11! + 10488970696371462*x^12/12! +...
%e A279844 Further, A( (2*A(x) + x)/3 ) = (A'(x) - 1)/(2*A'(x) + 1), which begins
%e A279844 A( (2*A(x) + x)/3 ) = x + 5*x^2/2! + 63*x^3/3! + 1311*x^4/4! + 38445*x^5/5! + 1464381*x^6/6! + 68939271*x^7/7! + 3879180855*x^8/8! + 254691006453*x^9/9! + 19160241768837*x^10/10! + 1628342402620383*x^11/11! + 154564849209408975*x^12/12! +...
%t A279844 m = 21; A[_] = 0;
%t A279844 Do[G[x_] = Integrate[A[x], x]; A[x_] = x + 3 G[(2 A[x] + x)/3] + O[x]^m // Normal, {m}];
%t A279844 CoefficientList[A[x], x]*Range[0, m - 1]! // Rest (* _Jean-François Alcover_, Oct 20 2019 *)
%o A279844 (PARI) /* A(x) = x + (p+q)*G((p*A(x) + q*x)/(p+q)) ; G(x) = Integral A(x) dx: */
%o A279844 {a(n, p=2, 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 A279844 for(n=1, 30, print1(a(n, 2, 1), ", "))
%o A279844 (PARI) /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
%o A279844 {a(n, p=2, 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 A279844 for(n=1, 30, print1(a(n, 2, 1), ", "))
%o A279844 (PARI) /* Informal code to generate the first N terms: */
%o A279844 {N=20; p=2; 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 A279844 Cf. A277410, A210949, A277403, A279843, A279845, A280570, A280571, A280572, A280573, A280574, A280575.
%K A279844 nonn
%O A279844 1,2
%A A279844 _Paul D. Hanna_, Dec 30 2016