A298511 Central Lehmer-Comtet numbers of the first kind: a(n) = A008296(2n,n).

1, 1, -1, -15, 49, 1365, -7645, -311311, 2475473, 132652377, -1367593305, -90881245455, 1151541572401, 91341008892445, -1373222414339685, -126594821384553375, 2202549127844351265, 231390624855674406705, -4573116447815658471025, -539278542630309415030735

Offset: 0

Author

Alois P. Heinz, Jan 20 2018

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386

Crossrefs

Cf. A008275, A008296, A048994.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=k, 1, `if`(k=0, 0,
      (n-1)*b(n-2, k-1)+b(n-1, k-1)+(k-n+1)*b(n-1, k)))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..25);

Mathematica

b[n_, k_] := b[n, k] = If[n == k, 1, If[k == 0, 0,
   (n-1) b[n-2, k-1] + b[n-1, k-1] + (k-n+1) b[n-1, k]]];
a[n_] := b[2n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 01 2023, from Maple code *)

Formula

a(n) = (2*n)!/n! * [x^(2*n)] ((1+x)*log(1+x))^n.

a(n) = Sum_{j=0..n} binomial(n+j,n) * n^j * Stirling1(2*n,n+j).