A068475 a(n) = Sum_{m=0..n} m*n^(m-1).

0, 1, 5, 34, 313, 3711, 54121, 937924, 18831569, 429794605, 10987654321, 310989720966, 9652968253897, 326011399456939, 11901025061692313, 466937872906120456, 19594541482740368161, 875711370981239308953, 41524755927216069067489, 2082225625247428808306410

Offset: 0

Author

Francois Jooste (phukraut(AT)hotmail.com), Mar 10 2002

Keywords

• nonn
• easy

Comments

The closed form comes from taking the derivative of the closed form of A031972, for which each term of this sequence is a derivative. - Jonas Whidden, Oct 18 2011

Examples

			a(2) = Sum_{m = 1..2} m*2^(m-1) = 1 + 2*2 = 5.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Crossrefs

Derivative sequence of A031972.

Cf. A023037, A062805, A062806, A368534.

Programs

Haskell

a068475 n = sum $ zipWith (*) [1..n] $ iterate (* n) 1
-- Reinhard Zumkeller, Nov 22 2014

Magma

[0] cat [(&+[m*n^(m-1): m in [0..n]]): n in [1..30]]; // G. C. Greubel, Oct 13 2018

Maple

a := n->sum(m*n^(m-1),m=1..n);

Mathematica

Join[{0}, Table[Sum[m*n^(m-1), {m,0,n}], {n,1,30}]] (* G. C. Greubel, Oct 13 2018 *)

PARI

for(n=0,30, print1(if(n==0, 0, sum(m=0,n, m*n^(m-1))), ", ")) \\ G. C. Greubel, Oct 13 2018

Formula

a(1) = 1. For n > 1, a(n) = ((n-1)*(n+1)*n^n - n^(n+1) + 1)/(n-1)^2. - Jonas Whidden, Oct 18 2011

a(n) = A062806(n) / n for n>=1. - Reinhard Zumkeller, Nov 22 2014

a(n) = [x^(n-1)] 1/((1 - x)*(1 - n*x)^2). - Peter Bala, Feb 12 2024