A390898 a(n) = Sum_{k>=0} (1/2)^(k+1) * |Stirling1(n+k,k)|.

1, 2, 20, 328, 7504, 220448, 7910720, 335381632, 16403174656, 909119017472, 56309497287680, 3854624637995008, 288982628489580544, 23548261603636748288, 2072327001908069089280, 195876746616926623793152, 19790823148886403832938496, 2128585470438970893454671872

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

G.f.: (1/2) * Sum_{k>=0} Product_{j=0..k-1} (1/2 + j*x).

a(0) = 1; a(n) = -2*a(n-1) + 2*Sum_{k=0..n-1} binomial(n+1,k+1) * a(k) * a(n-1-k).

a(n) = 2^n * A032188(n+1).

From Vaclav Kotesovec, Jan 20 2026: (Start)

a(n) ~ 2^(n-1) * n^n / (exp(n) * (1 - log(2))^(n + 1/2)).

E.g.f: 1/2 - 1/(2 + 2*LambertW(-1, -2*exp(-2 + 2*x))). (End)

Mathematica

a[0]=1;
a[n_]:=a[n]=-2 *a[n-1]+2 *Sum[Binomial[n+1, k+1] a[k]*a[n-1-k], {k, 0, n-1}];
aList[N_]:=Table[a[n], {n, 0, N}]; aList[17] (* Vincenzo Librandi, Nov 28 2025 *)
nmax = 20; Assuming[{x > 0}, CoefficientList[Series[1/2 - 1/(2 + 2*ProductLog[-1, -2*E^(-2 + 2*x)]), {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Jan 20 2026 *)

Prog

(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-2*v[i]+2*sum(j=0, i-1, binomial(i+1, j+1)*v[j+1]*v[i-j])); v;
(Magma) function aVector(N)
    v := [Integers()|1] cat [Integers()|0 : i in [1..N]];
    for n in [1..N] do v[n+1] := -2*v[n] + 2* &+[Binomial(n+1, k+1)*v[k+1]*v[n-k] : k in [0..n-1]];
end for; return v; end function; aVector(17); // Vincenzo Librandi, Nov 28 2025

Crossrefs

Cf. A390899, A390900.

Cf. A008275, A032188, A205671, A390890.

Sequence in context: A323574 A294454 A322729 * A304861 A104462 A352601

Adjacent sequences: A390895 A390896 A390897 * A390899 A390900 A390901

Keyword

nonn

Author

Seiichi Manyama, Nov 23 2025

Status

approved