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A279845 E.g.f. satisfies: A(x - Integral 2*A(x) dx) = x - Integral A(x) dx. 9
1, 1, 7, 87, 1577, 37921, 1143991, 41734167, 1793837945, 89100737537, 5038278258759, 320488252355991, 22712229678267017, 1778818548078114337, 152926844472960316055, 14348332105800041202903, 1461880180517958608890585, 161034066043430013259095681, 19105043857756090069661974951, 2432865068875486088572762200535, 331511875063241457659846364208233, 48205214775404458968179455649349921, 7461345443274460130807423699070922103 (list; graph; refs; listen; history; text; internal format)
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: A295527 A173812 A199027 * A231447 A020556 A303061

Adjacent sequences:  A279842 A279843 A279844 * A279846 A279847 A279848

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 05 2017

STATUS

approved

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Last modified August 5 22:41 EDT 2021. Contains 346488 sequences. (Running on oeis4.)