A319467 G.f. A(x) satisfies: Sum_{n>=0} x^n * ( 1/(1-x)^n - A(x) )^n = 1.

1, 1, 1, 2, 5, 17, 69, 310, 1530, 8079, 45325, 268362, 1667358, 10831443, 73328952, 515991884, 3765585347, 28445023673, 222041323198, 1788408340413, 14842961368603, 126784166379119, 1113305431684358, 10039694886817421, 92889877402814064, 880990917219289747, 8557926084570679399, 85078805258416836197, 864992386976470929447, 8987570189084296089971, 95373920468324819686521

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 17*x^5 + 69*x^6 + 310*x^7 + 1530*x^8 + 8079*x^9 + 45325*x^10 + 268362*x^11 + 1667358*x^12 + ...
where
1 = 1 + x*(1/(1-x) - A(x)) + x^2*(1/(1-x)^2 - A(x))^2 + x^3*(1/(1-x)^3 - A(x))^3 + x^4*(1/(1-x)^4 - A(x))^4 + x^5*(1/(1-x)^5 - A(x))^5 + ...
Also,
1 = 1/(1 + x*A(x)) + x*(1-x)/((1-x) + x*A(x))^2 + x^2*(1-x)^2/((1-x)^2 + x*A(x))^3 + x^3*(1-x)^3/((1-x)^3 + x*A(x))^4 + x^4*(1-x)^4/((1-x)^4 + x*A(x))^5 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=1, #A, x^m*(1/(1-x)^m - Ser(A))^m), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A325577.

Sequence in context: A372376 A303952 A162037 * A326412 A183239 A263639

Adjacent sequences: A319464 A319465 A319466 * A319468 A319469 A319470

Keyword

nonn

Author

Paul D. Hanna, Sep 29 2018

Status

approved