A280570 - OEIS

%I #15 Sep 30 2019 09:19:31

%S 1,4,28,332,5748,131940,3791692,131375324,5343640212,250142552212,

%T 13271217848604,788346022938556,51916178572447140,3759254932421361284,

%U 297243198474965188732,25513664852425377663756,2365246919693613357168916,235776253411115081902083556,25174157913006507920211300588,2869108641038261410331666767772

%N E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral 3*A(x) dx.

%F E.g.f. A(x) satisfies:

%F (1) A(x - Integral A(x) dx) = x + Integral 3*A(x) dx.

%F (2) A(x) = x + 4 * G( (A(x) + 3*x)/4 ), where G(x) = Integral A(x) dx.

%F (3) A(x) = -3*x + 4 * Series_Reversion(x - Integral A(x) dx).

%F (4) A( (A(x) + 3*x)/4 ) = (A'(x) - 1)/(A'(x) + 3).

%F (5) A'(x - Integral A(x) dx) = (1 + 3*A(x))/(1 - A(x)).

%F a(n) = Sum_{k=0..n} A277410(n,k) * 4^(n-k-1).

%e E.g.f.: A(x) = x + 4*x^2/2! + 28*x^3/3! + 332*x^4/4! + 5748*x^5/5! + 131940*x^6/6! + 3791692*x^7/7! + 131375324*x^8/8! + 5343640212*x^9/9! + 250142552212*x^10/10! + 13271217848604*x^11/11! + 788346022938556*x^12/12! +...

%e Let G(x) = Integral A(x) dx, then A(x - G(x)) = x + 3*G(x) where

%e G(x) = x^2/2! + 4*x^3/3! + 28*x^4/4! + 332*x^5/5! + 5748*x^6/6! + 131940*x^7/7! + 3791692*x^8/8! + 131375324*x^9/9! + 5343640212*x^10/10! + 250142552212*x^11/11! + 13271217848604*x^12/12! +...

%e Also, A(x) = x + 4 * G( (A(x) + 3*x)/4 ).

%e RELATED SERIES.

%e We have (A(x) + 3*x)/4 = Series_Reversion( x - Integral A(x) dx ), where

%e (A(x) + 3*x)/4 = x + x^2/2! + 7*x^3/3! + 83*x^4/4! + 1437*x^5/5! + 32985*x^6/6! + 947923*x^7/7! + 32843831*x^8/8! + 1335910053*x^9/9! + 62535638053*x^10/10! + 3317804462151*x^11/11! + 197086505734639*x^12/12! +...

%e Further, A( (A(x) + 3*x)/4 ) = (A'(x) - 1)/(A'(x) + 3), which begins

%e A( (A(x) + 3*x)/4 ) = x + 5*x^2/2! + 47*x^3/3! + 707*x^4/4! + 14825*x^5/5! + 401033*x^6/6! + 13340739*x^7/7! + 528281555*x^8/8! + 24323141773*x^9/9! + 1279128727141*x^10/10! + 75770789421947*x^11/11! + 4999463984999615*x^12 +...

%t m = 21; A[_] = 0;

%t Do[A[x_] = -3 x + 4 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=3) = 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, 3), ", "))

%o (PARI) /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */

%o {a(n, p=1, q=3) = 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, 3), ", "))

%o (PARI) /* Informal code to generate the first N terms: */

%o {N=20; p=1; q=3; 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, A280571, A280572, A280573, A280574, A280575.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 05 2017