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

%I #15 May 06 2023 06:56:37

%S 1,1,1,3,21,271,5547,163449,6515653,336521487,21811917243,

%T 1731102129033,164965649015727,18576187504063053,2439032446517056113,

%U 369203184490259386011,63808807042506278660325,12485211237616581137265679,2745317734648664454455184459

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

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

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

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

%e + 21*x^4*((1+x)^4 - 4^2*x*(1+x)^3 + 6^2*x^2*(1+x)^2 - 4^2*x^3*(1+x) + x^4)

%e + 271*x^5*((1+x)^5 - 5^2*x*(1+x)^4 + 10^2*x^2*(1+x)^3 - 10^2*x^3*(1+x)^2 + 5^2*x^4*(1+x) - x^5) + ...

%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)^(k-j)+x*O(x^n))), n))}

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

%Y Cf. A227821.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jul 31 2013