A024383 a(n) = s(1)*s(2)*...*s(n)*(1/s(1) - 1/s(2) + ... + c/s(n)), where c = (-1)^(n+1) and s(k) = 4*k - 3 for k = 1, 2, 3, ....

1, 4, 41, 488, 8881, 176556, 4622745, 128838480, 4403082465, 157917434580, 6659489632905, 292097166060600, 14653855170875025, 759940716395000700, 44202442040567948025, 2645857155729629066400, 175060715455871850866625

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Crossrefs

Cf. A024199, A024217, A024396, A007696, A181048.

Programs

Maple

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 4 else 4*a(n-1) + (4*n - 3)^2*a(n-2) end if; end:
seq(a(n), n = 0..20);

Mathematica

Table[Product[4*k - 3, {k, 1, n}] * Sum[(-1)^(k+1)/(4*k - 3), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jan 02 2020 *)

PARI

a(n) = prod(k=1, n, 4*k-3)*sum(k=1, n, (-1)^(k+1)/(4*k-3)); \\ Michel Marcus, Jul 06 2019

Formula

a(n) ~ (Pi^(3/2) + 2*sqrt(Pi)*log(1 + sqrt(2))) * 2^(2*n - 2) * n^(n - 1/4) / (Gamma(1/4) * exp(n)). - Vaclav Kotesovec, Jan 02 2020

From Peter Bala, Mar 21 2024: (Start)

a(n) = Product_{k = 0..n} (4*k + 1) * Sum_{k = 0..n} (-1)^k/(4*k + 1).

a(n) = 4*a(n-1) + (4*n - 3)^2*a(n-2) with a(0) = 1 and a(1) = 4.

b(n) := Product_{k = 0..n} (4*k + 1) = A007696(n+1) satisfies the same 3-term recurrence with b(0) = 1 and b(1) = 5, leading to the continued fraction expansion for the constant A181048 = Sum_{k >= 0} (-1)^k/(4*k + 1) = 1/(1 + 1^2/(4 + 5^2/(4 + 9^2/(4 + 13^2/(4 + ... ))))) due to Euler. (End)

Extensions

More terms from Sean A. Irvine, Jul 06 2019