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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.

%I A060946 #32 Jun 17 2022 16:34:32
%S A060946 1,3,12,76,701,8477,126126,2223278,45269999,1045269999,26982694600,
%T A060946 769991065288,24068076187769,817782849441913,30010708874832538,
%U A060946 1182932213481679514,49844124089148547995,2235755683827845079963,106363105981739086612804
%N 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.
%C A060946 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
%H A060946 Harry J. Smith, <a href="/A060946/b060946.txt">Table of n, a(n) for n = 1..100</a>
%F A060946 a(n) = Sum_{k=1..n} k^(k-1).
%F A060946 a(n) = Sum_{k=1..n} A000169(k). - _Jonathan Vos Post_, Feb 12 2010
%e A060946 a(3) = 12 because the matrix is: 1,1,1 1,2,4 1,3,9 and the trace is 1+2+9 = 12.
%e A060946 1 = 1^0; 3 = 1^0 + 2^1; 12 = 1^0 + 2^1 + 3^2; 76 = 1^0 + 2^1 + 3^2 + 4^3.
%p A060946 a:=n-> sum((j+1)^j, j=0..n-1): seq(a(n), n=1..25); # _Zerinvary Lajos_, Dec 17 2008
%t A060946 Table[Sum[i^(i-1), {i, n}], {n, 25}]
%o A060946 (PARI) { for (n=1, 100, write("b060946.txt", n, " ", sum(k=1, n, k^(k - 1))); ) } \\ _Harry J. Smith_, Jul 15 2009
%o A060946 (Magma) [(&+[j^(j-1): j in [1..n]]): n in [1..25]]; // _G. C. Greubel_, Apr 09 2021
%o A060946 (Sage) [sum(j^(j-1) for j in (1..n)) for n in (1..25)] # _G. C. Greubel_, Apr 09 2021
%o A060946 (Python)
%o A060946 from itertools import accumulate, count, islice
%o A060946 def A060946_gen(): # generator of terms
%o A060946     yield from accumulate((k**(k-1) for k in count(1)))
%o A060946 A060946_list = list(islice(A060946_gen(),20)) # _Chai Wah Wu_, Jun 17 2022
%Y A060946 Cf. A000178, A001923.
%Y A060946 Cf. A000169. - _Jonathan Vos Post_, Feb 12 2010
%K A060946 nonn
%O A060946 1,2
%A A060946 Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001
%E A060946 More terms from _Zak Seidov_, Jul 07 2003