A060946 Trace of Vandermonde matrix of numbers 1,2,...,n, i.e., the matrix A with A[i,j] = i^(j-1), 1 <= i <= n, 1 <= j <= n.

1, 3, 12, 76, 701, 8477, 126126, 2223278, 45269999, 1045269999, 26982694600, 769991065288, 24068076187769, 817782849441913, 30010708874832538, 1182932213481679514, 49844124089148547995, 2235755683827845079963, 106363105981739086612804

Offset: 1

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001

Keywords

• nonn

Comments

A member of the family of sequences defined by a(n) = Sum_{i=1..n}[i*c(1)*..*c(r)]^(i-1); c(i) integers. Here c(1)=1. - Ctibor O. Zizka, Feb 23 2008

Examples

			a(3) = 12 because the matrix is: 1,1,1 1,2,4 1,3,9 and the trace is 1+2+9 = 12.
1 = 1^0; 3 = 1^0 + 2^1; 12 = 1^0 + 2^1 + 3^2; 76 = 1^0 + 2^1 + 3^2 + 4^3.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000178, A001923.

Cf. A000169. - Jonathan Vos Post, Feb 12 2010

Programs

Magma

[(&+[j^(j-1): j in [1..n]]): n in [1..25]]; // G. C. Greubel, Apr 09 2021

Maple

a:=n-> sum((j+1)^j, j=0..n-1): seq(a(n), n=1..25); # Zerinvary Lajos, Dec 17 2008

Mathematica

Table[Sum[i^(i-1), {i, n}], {n, 25}]

PARI

{ for (n=1, 100, write("b060946.txt", n, " ", sum(k=1, n, k^(k - 1))); ) } \\ Harry J. Smith, Jul 15 2009

Python

from itertools import accumulate, count, islice
def A060946_gen(): # generator of terms
    yield from accumulate((k**(k-1) for k in count(1)))
A060946_list = list(islice(A060946_gen(),20)) # Chai Wah Wu, Jun 17 2022

Sage

[sum(j^(j-1) for j in (1..n)) for n in (1..25)] # G. C. Greubel, Apr 09 2021

Formula

a(n) = Sum_{k=1..n} k^(k-1).

a(n) = Sum_{k=1..n} A000169(k). - Jonathan Vos Post, Feb 12 2010

Extensions

More terms from Zak Seidov, Jul 07 2003