A059302 A diagonal of A008296.

-1, -1, 5, 25, 70, 154, 294, 510, 825, 1265, 1859, 2639, 3640, 4900, 6460, 8364, 10659, 13395, 16625, 20405, 24794, 29854, 35650, 42250, 49725, 58149, 67599, 78155, 89900, 102920, 117304, 133144, 150535, 169575, 190365, 213009, 237614, 264290, 293150, 324310, 357889, 394009, 432795, 474375

Offset: 2

Author

N. J. A. Sloane, Jan 26 2001

Keywords

• sign
• easy

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,n-2).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Crossrefs

Cf. similar sequences listed in A241765.

Programs

Maple

with(combinat): for n from 3 to 100 do for k from n-2 to n-2 do printf(`%d,`,sum(binomial(l,k)*k^(l-k)*stirling1(n,l), l=k..n)) od: od:

Mathematica

f[n_] := 3*n - 1; s1 = s2 = s3 = 0; lst = {}; Do[a = f[n]; s1 += a;
s2 += s1; s3 += s2; AppendTo[lst, s3], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 27 2009 *)
Drop[CoefficientList[Series[-x^2 (1 - 4*x)/(1 - x)^5, {x, 0, 50}], x], 2] (* Vincenzo Librandi, Mar 22 2012 *)
Rest[Table[(n - 1) n (n + 1)(3 n - 10)/24, {n, 50}]] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1},{-1, -1, 5, 25, 70}, 50] (* Harvey P. Dale, Jun 05 2012 *)

PARI

x='x+O('x^99); Vec(x^2*(-1+4*x)/(1-x)^5) \\ Altug Alkan, Sep 13 2017

Formula

a(n) = (n-1)n(n+1)(3n-10)/24.

If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then a(n-1) = f(n,n-2,-1), for n >= 3. - Milan Janjic, Dec 20 2008

G.f.: -x^2*(1-4*x)/(1-x)^5. - Colin Barker, Mar 21 2012

a(2)=-1, a(3)=-1, a(4)=5, a(5)=25, a(6)=70, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Jun 05 2012

a(n) = Sum_{k=1..n} Sum_{i=1..k} (n-i)*(n-k-1). - Wesley Ivan Hurt, Sep 12 2017

Extensions

More terms from James Sellers, Jan 26 2001