A319947 - OEIS

%I #14 Oct 10 2020 05:15:01

%S 1,2,17,233,4457,109599,3294200,117023348,4796944724,222859320409,

%T 11572143728964,664158801170094,41748985785588788,2852580634624308469,

%U 210503045435437702457,16684642612290860954017,1413651317086090261964496,127503642994522759923638691,12197174216389125259958117521,1233478106868364650369933771887

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

%C Compare to A319466, the dual to this sequence.

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

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

%H Paul D. Hanna, <a href="/A319947/b319947.txt">Table of n, a(n) for n = 0..300</a>

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

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

%F a(n) ~ c * d^n * n! / sqrt(n), where d = 5.466604933212768466569984392298244498368362826438280277089... and c = 0.42786673435712807571161365324459616568268597937553... - _Vaclav Kotesovec_, Oct 10 2020

%e 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 + ...

%e such that

%e 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  + ...

%e Equivalently,

%e A(x) = 1  +

%e (1/(1-x) - (1-x))  +

%e (1/(1-x)^4 - 2 + (1-x)^4)  +

%e (1/(1-x)^9 - 3/(1-x)^3 + 3*(1-x)^3 - (1-x)^9)  +

%e (1/(1-x)^16 - 4/(1-x)^8 + 6 - 4*(1-x)^8 + (1-x)^16)  +

%e (1/(1-x)^25 - 5/(1-x)^15 + 10/(1-x)^5 - 10*(1-x)^5 + 5*(1-x)^15 - (1-x)^25)  +

%e (1/(1-x)^36 - 6/(1-x)^24 + 15/(1-x)^12 - 20 + 15*(1-x)^12 - 6*(1-x)^24 + (1-x)^36)  +

%e ...

%o (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)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A319466.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 08 2018