A225602 -id:A225602 - cp's OEIS frontend

A350305 a(n) is the constant term in the expansion of Product_{k=1..n} (x^k + 1 + 1/x^k)^n.

1, 1, 13, 1437, 1884211, 24657701475, 3111336932350947, 3710920324904591897521, 41323213770479673319301068309, 4261037235228828189774620497534270303, 4045313784246510024420372971256850718016451185

Offset: 0

Seiichi Manyama, Dec 23 2021

nonn

a(n) is the coefficient of x^(n^2 * (n+1)/2) in Product_{k=0..n} (1 + x^k + x^(2*k))^n.

Robert Israel, Table of n, a(n) for n = 0..44

Cf. A007576, A156181, A225602, A350282, A350306, A350307.

f:= n -> coeff(mul(x^k+1+1/x^k,k=1..n)^n,x,0):
map(f, [$0..12]); # Robert Israel, Jan 15 2023

a[n_] := Coefficient[Series[Product[(x^k + 1 + 1/x^k)^n, {k, 1, n}], {x, 0, 0}], x, 0]; Array[a, 11, 0] (* Amiram Eldar, Dec 24 2021 *)

a(n) = polcoef(prod(k=1, n, x^k+1+1/x^k)^n, 0);

a(n) = polcoef(prod(k=1, n, 1+x^k+x^(2*k))^n, n^2*(n+1)/2);

A225604 G.f.: exp( Sum_{n>=1} A002426(n^2) * x^n/n ), where A002426 is the central trinomial coefficients.

Original entry on oeis.org

1, 1, 10, 1056, 1300253, 16436676927, 2026538428535847, 2377041996570919354629, 26137381916593225072659360863, 2668615348740645885804068311893052895, 2513426521807431879643802805359800329740903335, 21735453667359385540995804455408000917620356989063370267

Offset: 0

Paul D. Hanna, Aug 03 2013

nonn

			G.f.: A(x) = A(x) = 1 + x + 10*x^2 + 1056*x^3 + 1300253*x^4 + 16436676927*x^5 +...
where
log(A(x)) = x + 19*x^2/2 + 3139*x^3/3 + 5196627*x^4/4 + 82176836301*x^5/5 +...+ A225602(n)*x^n/n +...

Cf. A225602, A002426.

{A002426(n)=sum(k=0,n, binomial(n, k)*binomial(k, n-k))}
{a(n)=polcoeff(exp(sum(m=1,n+1,A002426(m^2)*x^m/m) +x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

Logarithmic derivative yields A225602.

cp's OEIS Frontend

A350305 a(n) is the constant term in the expansion of Product_{k=1..n} (x^k + 1 + 1/x^k)^n.

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A225604 G.f.: exp( Sum_{n>=1} A002426(n^2) * x^n/n ), where A002426 is the central trinomial coefficients.

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