A353258 - OEIS

%I #11 Apr 09 2022 08:48:48

%S 1,0,-1,-3,-17,-153,-1846,-27828,-503000,-10599873,-255143728,

%T -6906078108,-207627211745,-6864486246225,-247526246562328,

%U -9667515778323735,-406560434763167342,-18316445888374834635,-880110629723965618045,-44928348211160605056537

%N Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (3 * j - x).

%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k * 3^(n-2*k) * |Stirling1(n-k,k)|.

%t a[n_] := Sum[(-1)^k * 3^(n - 2*k) * Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}]; Array[a, 20, 0] (* _Amiram Eldar_, Apr 09 2022 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, 3*j-x)))

%o (PARI) a(n) = sum(k=0, n\2, (-1)^k*3^(n-2*k)*abs(stirling(n-k, k, 1)));

%Y Cf. A353255, A353256, A353257.

%K sign

%O 0,4

%A _Seiichi Manyama_, Apr 08 2022