A319947 G.f.: Sum_{n>=0} ( 1/(1-x)^n - (1-x)^n )^n.

1, 2, 17, 233, 4457, 109599, 3294200, 117023348, 4796944724, 222859320409, 11572143728964, 664158801170094, 41748985785588788, 2852580634624308469, 210503045435437702457, 16684642612290860954017, 1413651317086090261964496, 127503642994522759923638691, 12197174216389125259958117521, 1233478106868364650369933771887

Offset: 0

Author

Paul D. Hanna, Oct 08 2018

Keywords

• nonn

Comments

Compare to A319466, the dual to this sequence.

G.f. A(x) = (1-x) * B( x/(1-x) ), where B(x) is the g.f. of A319466.

a(n) - A319466(n) = 0 (mod 2) for n >= 0.

Examples

			G.f.: A(x) = 1 + 2*x + 17*x^2 + 233*x^3 + 4457*x^4 + 109599*x^5 + 3294200*x^6 + 117023348*x^7 + 4796944724*x^8 + 222859320409*x^9 + ...
such that
A(x) = 1  +  (1/(1-x) - (1-x))  +  (1/(1-x)^2 - (1-x)^2)^2  +  (1/(1-x)^3 - (1-x)^3)^3  +  (1/(1-x)^4 - (1-x)^4)^4  +  (1/(1-x)^5 - (1-x)^5)^5  + ...
Equivalently,
A(x) = 1  +
(1/(1-x) - (1-x))  +
(1/(1-x)^4 - 2 + (1-x)^4)  +
(1/(1-x)^9 - 3/(1-x)^3 + 3*(1-x)^3 - (1-x)^9)  +
(1/(1-x)^16 - 4/(1-x)^8 + 6 - 4*(1-x)^8 + (1-x)^16)  +
(1/(1-x)^25 - 5/(1-x)^15 + 10/(1-x)^5 - 10*(1-x)^5 + 5*(1-x)^15 - (1-x)^25)  +
(1/(1-x)^36 - 6/(1-x)^24 + 15/(1-x)^12 - 20 + 15*(1-x)^12 - 6*(1-x)^24 + (1-x)^36)  +
...
		

Links

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

Crossrefs

Cf. A319466.

Programs

PARI

{a(n) = my(A=1, X=x + x*O(x^n)); A = sum(m=0,n, (1/(1-X)^m - (1-x)^m)^m );polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} 1/(1-x)^(n^2) * Sum_{k=0..n} (-1)^k * binomial(n,k) * (1-x)^(2*n*k).

G.f.: Sum_{n>=0} (1-x)^(n^2) * Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) / (1-x)^(2*n*k).

a(n) ~ c * d^n * n! / sqrt(n), where d = 5.466604933212768466569984392298244498368362826438280277089... and c = 0.42786673435712807571161365324459616568268597937553... - Vaclav Kotesovec, Oct 10 2020