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

0,3

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

Links

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

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

Example

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

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 *)

Prog

(Haskell)
a068475 n = sum $ zipWith (*) [1..n] $ iterate (* n) 1
-- Reinhard Zumkeller, Nov 22 2014
(PARI) for(n=0, 30, print1(if(n==0, 0, sum(m=0, n, m*n^(m-1))), ", ")) \\ G. C. Greubel, Oct 13 2018
(Magma) [0] cat [(&+[m*n^(m-1): m in [0..n]]): n in [1..30]]; // G. C. Greubel, Oct 13 2018

Crossrefs

Derivative sequence of A031972.

Cf. A023037, A062805, A062806, A368534.

Sequence in context: A362912 A328488 A258179 * A097817 A344556 A303175

Adjacent sequences: A068472 A068473 A068474 * A068476 A068477 A068478

Keyword

nonn,easy

Author

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

Status

approved