0,3

For k >= 2, the difference a(n+k) - a(n) is divisible by k. It follows that for each k, the sequence formed by taking a(n) modulo k is periodic with period dividing k. For example, modulo 10 the sequence becomes [1, 1, 7, 9, 5, 1, 1, 7, 9, 5, ...], a purely periodic sequence of period 5. Cf. A047974. - Peter Bala, Feb 11 2025

Eric Weisstein's World of Mathematics, Bell Polynomial.

a(n) = Sum_{k=0..n} 4^(n-k) * |Stirling1(n,k)| * A004211(k) = 4^n * Sum_{k=0..n} (1/2)^k * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

From Vaclav Kotesovec, Aug 02 2024: (Start)

a(n) = 6*(2*n - 3)*a(n-1) - (48*n^2 - 192*n + 191)*a(n-2) + 32*(n-3)*(n-2)*(2*n - 5)*a(n-3).

a(n) ~ 2^(2*n - 1/6) * n^(n - 1/3) / (sqrt(3) * exp(n - 3*2^(-4/3)*n^(1/3) + 1/2)) * (1 - 31/(72*2^(2/3)*n^(1/3)) - 4607/(20736*2^(1/3)*n^(2/3))). (End)

a(n) = (1/exp(1/2)) * (-4)^n * n! * Sum_{k>=0} binomial(-k/2,n)/(2^k * k!). - Seiichi Manyama, Jan 18 2025

Table[4^n * Sum[Abs[StirlingS1[n, k]] * BellB[k, 1/2] / 2^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 02 2024 *)

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-4*x)^(1/2)-1)/2)))

Cf. A080893, A375175.

Cf. A004211.

Sequence in context: A381601 A112700 A365039 * A365782 A362773 A235370

Adjacent sequences: A375170 A375171 A375172 * A375174 A375175 A375176

nonn,easy

Seiichi Manyama, Aug 02 2024

approved