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G.f.: 1/(1-x) = Sum_{n>=0} a(n) * x^n/(1-x)^n * Sum_{k=0..n} binomial(n,k)^2 * (-x)^k.
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%I #8 May 04 2023 18:07:00

%S 1,1,1,3,23,319,6857,209259,8563855,451423559,29740026091,

%T 2391941092881,230478978551687,26197466746328951,3467374262207936333,

%U 528520864124393733623,91899269489447224280211,18078003975588275698610731,3994026796748854058413543011,984658830428133667413074092081

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

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

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

%e + 3*x^3*(1 - 3^2*x + 3^2*x^2 - x^3)/(1-x)^3

%e + 23*x^4*(1 - 4^2*x + 6^2*x^2 - 4^2*x^3 + x^4)/(1-x)^4

%e + 319*x^5*(1 - 5^2*x + 10^2*x^2 - 10^2*x^3 + 5^2*x^4 - x^5)/(1-x)^5

%e + 6857*x^6*(1 - 6^2*x + 15^2*x^2 - 20^2*x^3 + 15^2*x^4 - 6^2*x^5 + x^6)/(1-x)^6 +...

%o (PARI)

%o {a(n)=if(n==0, 1, 1-polcoeff(sum(k=0, n-1, a(k)*x^k*sum(j=0, k,binomial(k,j)^2*(-x)^j)/(1-x+x*O(x^n))^k), n))}

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

%Y Cf. A227820.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jul 31 2013