This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A279843 #25 Jan 05 2017 19:09:50
%S A279843 1,3,18,189,2907,59373,1520019,46964934,1705072680,71304998301,
%T A279843 3382510434561,179805942701262,10604941134768027,688310036217216666,
%U A279843 48823571643364894410,3762575719966561217301,313432935903428395412205,28098727418570995251538128,2700377607104440375587008499,277246288187233901613660728700
%N A279843 E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral 2*A(x) dx.
%H A279843 Paul D. Hanna, <a href="/A279843/b279843.txt">Table of n, a(n) for n = 1..150</a>
%F A279843 E.g.f. A(x) satisfies:
%F A279843 (1) A(x - Integral A(x) dx) = x + Integral 2*A(x) dx.
%F A279843 (2) A(x) = x + 3 * G( (A(x) + 2*x)/3 ), where G(x) = Integral A(x) dx.
%F A279843 (3) A(x) = -2*x + 3 * Series_Reversion(x - Integral A(x) dx).
%F A279843 (4) A( (A(x) + 2*x)/3 ) = (A'(x) - 1)/(A'(x) + 2).
%F A279843 (5) A'(x - Integral A(x) dx) = (1 + 2*A(x))/(1 - A(x)).
%F A279843 a(n) = Sum_{k=0..n-1} A277410(n,k) * 3^(n-k-1).
%e A279843 E.g.f.: A(x) = x + 3*x^2/2! + 18*x^3/3! + 189*x^4/4! + 2907*x^5/5! + 59373*x^6/6! + 1520019*x^7/7! + 46964934*x^8/8! + 1705072680*x^9/9! + 71304998301*x^10/10! + 3382510434561*x^11/11! + 179805942701262*x^12/12! +...
%e A279843 Let G(x) = Integral A(x) dx, then A(x - G(x)) = x + 2*G(x) where
%e A279843 G(x) = x^2/2! + 3*x^3/3! + 18*x^4/4! + 189*x^5/5! + 2907*x^6/6! + 59373*x^7/7! + 1520019*x^8/8! + 46964934*x^9/9! + 1705072680*x^10/10! +...
%e A279843 Also, A(x) = x + 3 * G( (A(x) + 2*x)/3 ).
%e A279843 RELATED SERIES.
%e A279843 We have (A(x) + 2*x)/3 = Series_Reversion( x - Integral A(x) dx ), where
%e A279843 (A(x) + 2*x)/3 = x + x^2/2! + 6*x^3/3! + 63*x^4/4! + 969*x^5/5! + 19791*x^6/6! + 506673*x^7/7! + 15654978*x^8/8! + 568357560*x^9/9! + 23768332767*x^10/10! + 1127503478187*x^11/11! + 59935314233754*x^12/12! +...
%e A279843 Further, A( (A(x) + 2*x)/3 ) = (A'(x) - 1)/(A'(x) + 2), which begins
%e A279843 A( (A(x) + 2*x)/3 ) = x + 4*x^2/2! + 33*x^3/3! + 441*x^4/4! + 8241*x^5/5! + 199071*x^6/6! + 5922360*x^7/7! + 209986506*x^8/8! + 8665824933*x^9/9! + 408861881955*x^10/10! + 21747689650404*x^11/11! +...
%o A279843 (PARI) /* A(x) = x + (p+q)*G((p*A(x) + q*x)/(p+q)) ; G(x) = Integral A(x) dx: */
%o A279843 {a(n,p=1,q=2) = 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 A279843 for(n=1, 30, print1(a(n,1,2), ", "))
%o A279843 (PARI) /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
%o A279843 {a(n,p=1,q=2) = 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 A279843 for(n=1, 30, print1(a(n,1,2), ", "))
%o A279843 (PARI) /* Informal code to generate the first N terms: */
%o A279843 {N=20; p=1;q=2; 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 A279843 Cf. A277410, A210949, A277403, A279844, A280571, A280572, A280573, A280574, A280575.
%K A279843 nonn
%O A279843 1,2
%A A279843 _Paul D. Hanna_, Dec 29 2016