A325154 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(2*n-1) - A(x))^n.

1, 1, 4, 60, 1349, 40210, 1470027, 63225750, 3116555468, 172936040306, 10661699020596, 722933543336296, 53476543241702021, 4286318739039468220, 370139507278333619231, 34264675353237245461705, 3385595826616475280589858, 355676742010175185149150523, 39592541401227701053287450374, 4655516336942715288212969823798, 576645913391345319618489456738288, 75048370900002385430200781452328814

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = 1 + x + 4*x^2 + 60*x^3 + 1349*x^4 + 40210*x^5 + 1470027*x^6 + 63225750*x^7 + 3116555468*x^8 + 172936040306*x^9 + 10661699020596*x^10 + ...
such that
1 = 1  +  ((1+x) - A(x))  +  ((1+x)^3 - A(x))^2  +  ((1+x)^5 - A(x))^3  +  ((1+x)^7 - A(x))^4  +  ((1+x)^9 - A(x))^5  +  ((1+x)^11 - A(x))^6  +  ((1+x)^13 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)/(1 + (1+x)^2*A(x))^2  +  (1+x)^6/(1 + (1+x)^4*A(x))^3  +  (1+x)^15/(1 + (1+x)^6*A(x))^4  +  (1+x)^28/(1 + (1+x)^8*A(x))^5  +  (1+x)^45/(1 + (1+x)^10*A(x))^6  +  (1+x)^66/(1 + (1+x)^6*A(x))^7 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(2*m-1) - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A303056.

Sequence in context: A181418 A208890 A370498 * A013486 A013483 A013484

Adjacent sequences: A325151 A325152 A325153 * A325155 A325156 A325157

Keyword

nonn

Author

Paul D. Hanna, Apr 12 2019

Status

approved