A325577 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1+x)^n - A(x))^n, where A(0) = 1.

1, 1, 0, 1, 2, 7, 30, 118, 581, 2858, 15252, 86277, 509438, 3167161, 20523646, 138503668, 971523468, 7063593311, 53158580100, 413391215663, 3316949183764, 27425390212610, 233382219349033, 2041727280750829, 18344048391099758, 169098977153523755, 1597886134800471284, 15464862942202804152, 153179560268616743995, 1551637987838963093709, 16062545697171306626153

Offset

0,5

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = 1 + x + x^3 + 2*x^4 + 7*x^5 + 30*x^6 + 118*x^7 + 581*x^8 + 2858*x^9 + 15252*x^10 + 86277*x^11 + 509438*x^12 + 3167161*x^13 + 20523646*x^14 + ...
such that
1 = 1 + x*((1+x) - A(x)) + x^2*((1+x)^2 - A(x))^2 + x^3*((1+x)^3 - A(x))^3 + x^4*((1+x)^4 - A(x))^4 + x^5*((1+x)^5 - A(x))^5 + x^6*((1+x)^6 - A(x))^6 + x^7*((1+x)^7 - A(x))^7 + x^8*((1+x)^8 - A(x))^8 + ...
also,
1 = 1/(1 + x*A(x)) + x*(1+x)/(1 + x*A(x)*(1+x))^2 + x^2*(1+x)^4/(1 + x*A(x)*(1+x)^2)^3 + x^3*(1+x)^9/(1 + x*A(x)*(1+x)^3)^4 + x^4*(1+x)^16/(1 + x*A(x)*(1+x)^4)^5 + x^5*(1+x)^25/(1 + x*A(x)*(1+x)^5)^6 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, x^m*((1+x)^m - Ser(A))^m), #A); ); A[n+1]}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A325575, A307940, A319467.

Sequence in context: A041805 A173233 A074416 * A097924 A027136 A353288

Adjacent sequences: A325574 A325575 A325576 * A325578 A325579 A325580

Keyword

nonn

Author

Paul D. Hanna, May 21 2019

Status

approved