A280573 E.g.f. satisfies: A(x - Integral 2*A(x) dx) = x + Integral 3*A(x) dx.

1, 5, 55, 1075, 30825, 1174725, 56153575, 3241453075, 219981653625, 17205716877125, 1527315775776375, 152004555650445875, 16793815038459239625, 2042866310966722613125, 271723598687954810434375, 39287423162026628955721875, 6143464129092882413626065625, 1034396495380447234136660853125, 186805274512176503194633726284375, 36060209533917578045193572845421875

Offset

1,2

Links

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

Formula

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

(1) A(x - Integral 2*A(x) dx) = x + Integral 3*A(x) dx.

(2) A(x) = x + 5 * G( (2*A(x) + 3*x)/5 ), where G(x) = Integral A(x) dx.

(3) A(x) = -3*x/2 + 5/2 * Series_Reversion(x - Integral 2*A(x) dx).

(4) A( (2*A(x) + 3*x)/5 ) = (A'(x) - 1)/(2*A'(x) + 3).

(5) A'(x - Integral 2*A(x) dx) = (1 + 3*A(x))/(1 - 2*A(x)).

a(n) = Sum_{k=0..n-1} A277410(n,k) * 2^k * 5^(n-k-1).

Example

E.g.f.: A(x) = x + 5*x^2/2! + 55*x^3/3! + 1075*x^4/4! + 30825*x^5/5! + 1174725*x^6/6! + 56153575*x^7/7! + 3241453075*x^8/8! + 219981653625*x^9/9! + 17205716877125*x^10/10! + 1527315775776375*x^11/11! + 152004555650445875*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - 2*G(x)) = x + 3*G(x) where
G(x) = x^2/2! + 5*x^3/3! + 55*x^4/4! + 1075*x^5/5! + 30825*x^6/6! + 1174725*x^7/7! + 56153575*x^8/8! + 3241453075*x^9/9! + 219981653625*x^10/10! + 17205716877125*x^11/11! + 1527315775776375*x^12/12! +...
Also, A(x) = x + 5 * G( (2*A(x) + 3*x)/5 ).
RELATED SERIES.
We have (2*A(x) + 3*x)/5 = Series_Reversion( x - Integral 2*A(x) dx ), where
(2*A(x) + 3*x)/5 =  x + 2*x^2/2! + 22*x^3/3! + 430*x^4/4! + 12330*x^5/5! + 469890*x^6/6! + 22461430*x^7/7! + 1296581230*x^8/8! + 87992661450*x^9 + 6882286750850*x^10 + 610926310310550*x^11 + 60801822260178350*x^12 +...
Further, A( (2*A(x) + 3*x)/5 ) = (A'(x) - 1)/(2*A'(x) + 3), which begins
A( (2*A(x) + 3*x)/5 ) = x + 7*x^2/2! + 107*x^3/3! + 2665*x^4/4! + 93005*x^5/5! + 4201015*x^6/6! + 233920155*x^7/7! + 15535390105*x^8/8! + 1201670102125*x^9/9! + 106329616511975*x^10/10! + 10612821894707675*x^11/11! + 1181462628283585225*x^12/12! +...

Mathematica

m = 21; A[_] = 0;
Do[A[x_] = -3x/2 + 5/2 InverseSeries[x-Integrate[2A[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=2, 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)}
for(n=1, 30, print1(a(n, 2, 3), ", "))
(PARI) /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=2, 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]}
for(n=1, 30, print1(a(n, 2, 3), ", "))
(PARI) /* Informal code to generate the first N terms: */
{N=20; p=2; 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))}

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279844, A279845, A280571, A280572, A280574, A280575.

Sequence in context: A119399 A177557 A158690 * A192723 A102221 A056600

Adjacent sequences: A280570 A280571 A280572 * A280574 A280575 A280576

Keyword

nonn

Author

Paul D. Hanna, Jan 05 2017

Status

approved