A280575 E.g.f. satisfies: A(x - Integral 4*A(x) dx) = x + Integral A(x) dx.

1, 5, 85, 2625, 119225, 7209725, 547774525, 50342086425, 5448583486625, 680816992367125, 96720369562897125, 15433474653279056625, 2738894920523846767625, 536165521694896664958125, 114986014404299081313978125, 26857525461337375682549015625, 6797751676147997291181732240625, 1856223312322488721077123869053125, 544728880837527263669006279810003125, 171209466763535665605709125529165390625

Offset

1,2

Links

Table of n, a(n) for n=1..20.

Formula

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

(1) A(x - Integral 4*A(x) dx) = x + Integral A(x) dx.

(2) A(x) = x + 5 * G( (4*A(x) + x)/5 ), where G(x) = Integral A(x) dx.

(3) A(x) = -x/4 + 5/4 * Series_Reversion(x - Integral 4*A(x) dx).

(4) A( (4*A(x) + x)/5 ) = (A'(x) - 1)/(4*A'(x) + 1).

(5) A'(x - Integral 4*A(x) dx) = (1 + A(x))/(1 - 4*A(x)).

a(n) = Sum_{k=0..n-1} A277410(n,k) * 4^k * 5^(n-k-1).

Example

E.g.f.: A(x) = x + 5*x^2/2! + 85*x^3/3! + 2625*x^4/4! + 119225*x^5/5! + 7209725*x^6/6! + 547774525*x^7/7! + 50342086425*x^8/8! + 5448583486625*x^9/9! + 680816992367125*x^10/10! + 96720369562897125*x^11/11! + 15433474653279056625*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - 4*G(x)) = x + G(x) where
G(x) = x^2/2! + 5*x^3/3! + 85*x^4/4! + 2625*x^5/5! + 119225*x^6/6! + 7209725*x^7/7! + 547774525*x^8/8! + 50342086425*x^9/9! + 5448583486625*x^10/10! + 680816992367125*x^11/11! + 96720369562897125*x^12/12! +...
Also, A(x) = x + 5 * G( (4*A(x) + x)/5 ).
RELATED SERIES.
We have (4*A(x) + x)/5 = Series_Reversion( x - Integral 4*A(x) dx ), where
(4*A(x) + x)/5 = x + 4*x^2/2! + 68*x^3/3! + 2100*x^4/4! + 95380*x^5/5! + 5767780*x^6/6! + 438219620*x^7/7! + 40273669140*x^8/8! + 4358866789300*x^9/9! + 544653593893700*x^10/10! + 77376295650317700*x^11/11! + 12346779722623245300*x^12/12! +...
Further, A( (4*A(x) + x)/5 ) = (A'(x) - 1)/(4*A'(x) + 1), which begins
A( (4*A(x) + x)/5 ) = x + 9*x^2/2! + 213*x^3/3! + 8365*x^4/4! + 463905*x^5/5! + 33459905*x^6/6! + 2985550445*x^7/7! + 318647659765*x^8/8! + 39707695580825*x^9/9! + 5672802966674825*x^10/10! + 916011286267596325*x^11/11! + 165283895506659660925*x^12/12! +...

Mathematica

m = 21; A[_] = 0;
Do[A[x_] = -x/4 + 5/4 InverseSeries[x-Integrate[4A[x], x] + O[x]^m], {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! // Rest (* Jean-François Alcover, Sep 30 2019 *)

Prog

(PARI) /* A(x) = x + (p+q)*G((p*A(x) + q*x)/(p+q)) ; G(x) = Integral A(x) dx: */
{a(n, p=4, 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)}
for(n=1, 30, print1(a(n, 4, 1), ", "))
(PARI) /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=4, 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]}
for(n=1, 30, print1(a(n, 4, 1), ", "))
(PARI) /* Informal code to generate the first N terms: */
{N=20; p=4; 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))}

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279844, A279845, A280571, A280572, A280573, A280574.

Sequence in context: A218139 A241330 A301435 * A318635 A203800 A277354

Adjacent sequences: A280572 A280573 A280574 * A280576 A280577 A280578

Keyword

nonn

Author

Paul D. Hanna, Jan 05 2017

Status

approved