A185826 Sum of the next n natural numbers raised to the n-th power.

1, 25, 3375, 1336336, 1160290625, 1870414552161, 5026507568359375, 20882706457600000000, 126834469112674289266929, 1078732544346879404306640625, 12415028528548173886807771291871, 188031682201497672618081000000000000, 3661926425131437024691115607984619140625

Offset: 1

Author

Amir H. Farrahi, Feb 05 2011

Keywords

• nonn

Comments

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.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..158

Crossrefs

Cf. A006003.

Programs

Mathematica

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 *)

Python

num = 100
n = 0
a = []
for i in range(1, num):
    sum = 0
    for j in range(1, i+1):
        sum = sum + (n+j)
    n = n + i
    a.append(sum**i)

Formula

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

a(n) = ((n + n^3)/2)^n. - Harvey P. Dale, Apr 24 2022

Extensions

Edited by N. J. A. Sloane, Feb 05 2011