A355154 G.f. A(x) satisfies: 1 = Sum_{n=-oo..+oo} (-x)^((n+1)^2) * ((1+x)^n - A(x))^n.

1, 3, 10, 47, 248, 1354, 7832, 46672, 285136, 1775728, 11232028, 71959338, 465981585, 3045116666, 20055877798, 132995908915, 887220714352, 5950074234287, 40092205226021, 271289342487845, 1842724189708458, 12559944025175681, 85877738644538351

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..400

Formula

G.f. A(x) satisfies:

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

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

(3) 1/(1+x) = Sum_{n=-oo..+oo} (-1)^n * (x + x^2)^(n^2) * (1+x)^(2*n) / (1 - A(x)*(1+x)^(n+1))^(n+1).

a(n) ~ c * d^n / n^(3/2), where d = 7.312110954492511257173117... and c = 0.1256708322688258093501... - Vaclav Kotesovec, Jun 22 2022

Example

G.f.: A(x) = x + 3*x^2 + 10*x^3 + 47*x^4 + 248*x^5 + 1354*x^6 + 7832*x^7 + 46672*x^8 + 285136*x^9 + 1775728*x^10 + 11232028*x^11 + ...
where
1 = ... + x^16/(1/(1+x)^5 - A(x))^5 - x^9/(1/(1+x)^4 - A(x))^4 + x^4/(1/(1+x)^3 - A(x))^3 - x/(1/(1+x)^2 - A(x))^2 + 1/(1/(1+x) - A(x)) - x + x^4*((1+x) - A(x)) - x^9*((1+x)^2 - A(x))^2 + x^16*((1+x)^3 - A(x))^3 -+ ...

Prog

(PARI) {a(n) = my(A=[0, 1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(n+4));
A[#A] = -polcoeff( sum(n=-t, t, (-x)^((n+1)^2) * ((1+x)^n - Ser(A))^n ), #A-1)); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A005651 A346055 A249479 * A236410 A339836 A105748

Adjacent sequences: A355151 A355152 A355153 * A355155 A355156 A355157

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2022

Status

approved