A062815 -id:A062815 - cp's OEIS frontend

A001923 a(n) = Sum_{k=1..n} k^k.

0, 1, 5, 32, 288, 3413, 50069, 873612, 17650828, 405071317, 10405071317, 295716741928, 9211817190184, 312086923782437, 11424093749340453, 449317984130199828, 18896062057839751444, 846136323944176515621

Offset: 0

N. J. A. Sloane

Starting from the second term, 1, the terms could be described as the special case (n=1; j=1) of the following general formula: a(n) =  Sum [(n + k - 1)]^(k) n=1; j=1; i=1,2,3,...,... For (n=0; j=1) the formula yields A062815 n=0; j=1; i=2,3,4,... For (n=2; j=0) we get A060946 and for (n=3; j=0) A117887. - Alexander R. Povolotsky, Sep 01 2007
From Luan Alberto Ferreira, Aug 01 2017: (Start)
If n == 0 or 3 (mod 4), then a(n) == 0 (mod 4).
If n == 0, 4, 7, 14, 15 or 17 (mod 18), then a(n) == 0 (mod 3). (End)
Called the hypertriangular function by M. K. Azarian. - Light Ediand, Nov 19 2021

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, p. 308.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
Mohammad K. Azarian, On the hyperfactorial function, hypertriangular function and the discriminants of certain polynomials, Int. J. Pure Appl. Math., Vol. 36, No. 2 (2007), pp. 251-257.
Andrew Cusumano, Problem H-656, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 45, No. 2 (2007), p. 187; A Sequence Tending To e, Solution to Problem H-656, ibid., Vol. 46-47, No. 3 (2008/2009), pp. 285-287.
G. W. Wishard (proposer) and F. Underwood (solution), Problem 4155: Bound for a Finite Sum, Amer. Math. Monthly, Vol. 53, No. 8 (1946), pp. 471-473.

Cf. A073825, A062970 (another version).
Cf. A062815, A060946, A117887.
Cf. A000312, A002109.

a001923 n = a001923_list !! n
a001923_list = scanl (+) 0 $ tail a000312_list
-- Reinhard Zumkeller, Jul 11 2014

Accumulate[Join[{0},Table[k^k,{k,20}]]] (* Harvey P. Dale, Feb 11 2015 *)

for(n=1,20,print1(sum(x=1,n,x^x), ", ")) \\ Jorge Coveiro, Dec 24 2004

# generates initial segment of sequence
from itertools import accumulate
def f(k): return 0 if k == 0 else k**k
def aupton(nn): return list(accumulate(f(k) for k in range(nn+1)))
print(aupton(17)) # Michael S. Branicky, Feb 12 2022

a(n) = A062970(n) - 1.
a(n+1)/a(n) > e*n and a(n+1)/a(n) is asymptotic to e*n. - Benoit Cloitre, Sep 29 2002
For n > 0: a(n) = a(n-1) + A000312(n). - Reinhard Zumkeller, Jul 11 2014
Limit_{n->oo} (a(n+2)/a(n+1) - a(n+1)/a(n)) = e (Cusumano, 2007). - Amiram Eldar, Jan 03 2022

A306377 a(n) = n^(n+1) - Sum_{k=1..n-1} k^(k+1).

Original entry on oeis.org

1, 7, 72, 934, 14511, 263197, 5468126, 128156252, 3346505197, 96372936395, 3034801313116, 103751149938746, 3827141124879891, 151520483911293537, 6408792648508559714, 288419881116435604320, 13761208522825454943617, 693870384974027681319231

Offset: 1

Kritsada Moomuang, Feb 11 2019

nonn

			a(5) = 5^6 - 4^5 - 3^4 - 2^3 - 1^2 = 14511.

Cf. A007778, A062815.

a(n) = n^(n+1) - sum(k=1, n-1, k^(k+1)); \\ Michel Marcus, Feb 11 2019

a(n) = 2*A007778(n) - A062815(n).

A341330 a(n) = Sum_{k=1..n} (-k)^(k+1).

Original entry on oeis.org

1, -7, 74, -950, 14675, -265261, 5499540, -128718188, 3358066213, -96641933787, 3041786442934, -103951418936138, 3833424966763151, -151734670591049073, 6416673685121841552, -288731231494230984304, 13774353220573494006705, -694460992134764182350927

Offset: 1

John H. Chakkour, Feb 09 2021

sign

Cf. A007778, A062815.

Mathematica
```
Accumulate[(-#)^(#+1)&/@Range[17]]
```
PARI
```
a(n) = sum(i=1, n, (-i)^(i+1));
```

sum = 0
for i in range(1,20):
    sum += (-i)**(i+1)
    print(sum, end = ", ")

cp's OEIS Frontend

A001923 a(n) = Sum_{k=1..n} k^k.

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A306377 a(n) = n^(n+1) - Sum_{k=1..n-1} k^(k+1).

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Crossrefs

Programs

PARI

Formula

A341330 a(n) = Sum_{k=1..n} (-k)^(k+1).

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Mathematica

PARI

Python