A352853 G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^n).

1, 1, 1, 2, 5, 15, 53, 205, 865, 3928, 18943, 96387, 514745, 2871568, 16670197, 100400979, 625756254, 4026925835, 26705001158, 182188059474, 1276736262332, 9178023547010, 67597658864028, 509525556949153, 3926577535219879, 30908466065826275, 248308190295151020

Offset

0,4

Links

Table of n, a(n) for n=0..26.

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^n),

(2) 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+1)).

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 53*x^6 + 205*x^7 + 865*x^8 + 3928*x^9 + 18943*x^10 + 96387*x^11 + 514745*x^12 + ...
where
(1) 1 = A(x) - x*A(x)*A(x*A(x)) + x^2*A(x)^2*A(x*A(x)^2) - x^3*A(x)^3*A(x*A(x)^3) + x^4*A(x)^4*A(x*A(x)^4) - x^5*A(x)^5*A(x*A(x)^5) + x^6*A(x)^6*A(x*A(x)^6) + ...
(2) 1 = 1/(1 + x*A(x)) + 1*x/(1 + x*A(x)^2) + 1*x^2/(1 + x*A(x)^3) + 2*x^3/(1 + x*A(x)^4) + 5*x^4/(1 + x*A(x)^5) + 15*x^5/(1 + x*A(x)^6) + 53*x^6/(1 + x*A(x)^7) + ... + a(n)*x^n/(1 + x*A(x)^(n+1)) + ...

Prog

(PARI) /* 1 = Sum_{n>=0} (-x)^n * A(x)^n * A(x*A(x)^n) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A-1, (-x)^n*Ser(A)^n*subst(Ser(A), x, x*Ser(A)^n) ), #A-1)); A[n+1]}
for(n=0, 31, print1(a(n), ", "))
(PARI) /* 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+1)) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A-1, A[n+1]*x^n/(1 + x*Ser(A)^(n+1)) ), #A-1)); A[n+1]}
for(n=0, 31, print1(a(n), ", "))

Crossrefs

Cf. A352854, A352855, A352856.

Sequence in context: A134381 A107589 A249892 * A006790 A007548 A360052

Adjacent sequences: A352850 A352851 A352852 * A352854 A352855 A352856

Keyword

nonn

Author

Paul D. Hanna, Apr 05 2022

Status

approved