A322725 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * (2 + x*A(x)^n)^n.

1, 2, 5, 14, 50, 220, 1107, 6030, 34643, 207704, 1293190, 8332942, 55406884, 379151494, 2664359328, 19193917324, 141571867121, 1068156408852, 8238449274801, 64921172524532, 522489723684089, 4293039694194962, 36000331681298631, 308011504511924924, 2687885268655409430, 23916543285143972648, 216912090405180557549

Offset

0,2

Links

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

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 50*x^4 + 220*x^5 + 1107*x^6 + 6030*x^7 + 34643*x^8 + 207704*x^9 + 1293190*x^10 + ...
such that
A(x) = 1 + x*(2 + x*A(x)) + x^2*(2 + x*A(x)^2)^2 + x^3*(2 + x*A(x)^3)^3 + x^4*(2 + x*A(x)^4)^4 + x^5*(2 + x*A(x)^5)^5 + x^6*(2 + x*A(x)^6)^6 + ...
Also, the g.f. satisfies the identity:
A(x) = 1/(1 - 2*x) + x^2*A(x)/(1 - 2*x*A(x))^2 + x^4*A(x)^4/(1 - 2*x*A(x)^2)^3 + x^6*A(x)^9/(1 - 2*x*A(x)^3)^4 + x^8*A(x)^16/(1 - 2*x*A(x)^4)^5 + x^10*A(x)^25/(1 - 2*x*A(x)^5)^6 + ...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*(2 + x*(A+x*O(x^n))^m)^m)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, x^(2*k)*A^(k^2)/(1 - 2*x*A^k +x*O(x^n))^(k+1) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A186998, A203014, A300049.

Sequence in context: A022562 A341360 A320954 * A339832 A245883 A115340

Adjacent sequences: A322722 A322723 A322724 * A322726 A322727 A322728

Keyword

nonn

Author

Paul D. Hanna, Jan 30 2019

Status

approved