A101851 a(n) = Sum_{k=0..n} (-1)^(n-k)*k*Stirling2(n,k).

0, 1, 1, -2, -1, 11, -18, -41, 317, -680, -1767, 19911, -68264, -59643, 2076973, -11905466, 18577387, 269836343, -2819431570, 12357816867, 17355428041, -752675321800, 6318046208653, -21416130683133, -152569023028272, 3016508107668601, -23667435182395287

Offset

0,4

Links

Robert Israel, Table of n, a(n) for n = 0..580

Formula

E.g.f.: (1-exp(-x))*exp(1-exp(-x)): G.f.: Sum(k*x^k/Product(1+l*x, l = 1 .. k), k = 1 .. infinity).

a(n) = Sum_{k=0..n} (-1)^(k+1)*binomial(n,k)*A000587(k+2). - Peter Luschny, Apr 17 2011

G.f.: x*G(0)/(1+x) where G(k) = 1 + 2*x*(k+1)/((2*k+1)*(2*x*k+2*x+1) - x*(2*k+1)*(2*k+3)*(2*x*k+2*x+1)/(x*(2*k+3) + 2*(k+1)*(2*x*k+3*x+1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012

G.f.: 1/x - G(0)/x, where G(k) = 1 - x^2*(k+1)/(x^2*(k+1) + (x*k + 1 - x)*(x*k + 1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 06 2014

a(n) = (-1)^n*(A000587(n)+A000587(n+1)). - Vladimir Reshetnikov, Oct 21 2015

Maple

A101851 := proc(n) local k;
add((-1)^(n-k)*k*combinat[stirling2](n, k), k = 0..n) end:
seq(A101851(n), n = 0..26); # Peter Luschny, Apr 17 2011

Mathematica

Table[Sum[(-1)^(n-k) k StirlingS2[n, k], {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Aug 09 2013 *)
Table[(-1)^n (BellB[n, -1] + BellB[n + 1, -1]), {n, 0, 25}] (* Vladimir Reshetnikov, Oct 21 2015 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*k*stirling(n, k, 2)); \\ Michel Marcus, Oct 22 2015

Crossrefs

Cf. A005493, A000587.

Sequence in context: A138351 A120293 A063624 * A300455 A270264 A305877

Adjacent sequences: A101848 A101849 A101850 * A101852 A101853 A101854

Keyword

easy,sign

Author

Vladeta Jovovic, Jan 27 2005

Status

approved