A279845 E.g.f. satisfies: A(x - Integral 2*A(x) dx) = x - Integral A(x) dx.

1, 1, 7, 87, 1577, 37921, 1143991, 41734167, 1793837945, 89100737537, 5038278258759, 320488252355991, 22712229678267017, 1778818548078114337, 152926844472960316055, 14348332105800041202903, 1461880180517958608890585, 161034066043430013259095681, 19105043857756090069661974951, 2432865068875486088572762200535, 331511875063241457659846364208233, 48205214775404458968179455649349921, 7461345443274460130807423699070922103

Offset

1,3

Links

Table of n, a(n) for n=1..23.

Formula

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

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

(2) A(x) = x + G( 2*A(x) - x ), where G(x) = Integral A(x) dx.

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

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

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

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

Example

E.g.f.: A(x) = x + x^2/2! + 7*x^3/3! + 87*x^4/4! + 1577*x^5/5! + 37921*x^6/6! + 1143991*x^7/7! + 41734167*x^8/8! + 1793837945*x^9/9! + 89100737537*x^10/10! + 5038278258759*x^11/11! + 320488252355991*x^12/12! + 22712229678267017*x^13/13! + 1778818548078114337*x^14/14! + 152926844472960316055*x^15/15! +...

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=-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, 2, -1), ", "))
(PARI) /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{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]}
for(n=1, 30, print1(a(n, 2, -1), ", "))
(PARI) /* Informal code to generate the first N terms: */
{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))}

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279844, A280570, A280571, A280572, A280573, A280574, A280575.

Sequence in context: A173812 A199027 A360237 * A231447 A020556 A303061

Adjacent sequences: A279842 A279843 A279844 * A279846 A279847 A279848

Keyword

nonn

Author

Paul D. Hanna, Jan 05 2017

Status

approved