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Sum of the next n natural numbers raised to the n-th power.
2

%I #24 Oct 26 2024 21:17:42

%S 1,25,3375,1336336,1160290625,1870414552161,5026507568359375,

%T 20882706457600000000,126834469112674289266929,

%U 1078732544346879404306640625,12415028528548173886807771291871,188031682201497672618081000000000000,3661926425131437024691115607984619140625

%N Sum of the next n natural numbers raised to the n-th power.

%C Write the natural numbers in groups, with group sizes incremented by one, each time: 1; 2,3; 4,5,6; 7,8,9,10; ... and add the numbers in each group, then take the i-th power for the i-th group to get the i-th entry in the sequence.

%H Harvey P. Dale, <a href="/A185826/b185826.txt">Table of n, a(n) for n = 1..158</a>

%F a(n) = A006003(n)^n.

%F a(n) = ((n + n^3)/2)^n. - _Harvey P. Dale_, Apr 24 2022

%t Module[{nn=15},#[[1]]^#[[2]]&/@Thread[{Total/@TakeList[Range[(nn(nn+1))/2],Range[ nn]],Range[nn]}]] (* or *) Table[((n+n^3)/2)^n,{n,20}] (* _Harvey P. Dale_, Apr 24 2022 *)

%o (Python)

%o num = 100

%o n = 0

%o a = []

%o for i in range(1, num):

%o sum = 0

%o for j in range(1, i+1):

%o sum = sum + (n+j)

%o n = n + i

%o a.append(sum**i)

%Y Cf. A006003.

%K nonn

%O 1,2

%A _Amir H. Farrahi_, Feb 05 2011

%E Edited by _N. J. A. Sloane_, Feb 05 2011