A123094 Sum of first n 12th powers.

0, 1, 4097, 535538, 17312754, 261453379, 2438235715, 16279522916, 84998999652, 367428536133, 1367428536133, 4505856912854, 13421957361110, 36720042483591, 93413954858887, 223160292749512, 504635269460168, 1087257506689929, 2244088888116105, 4457403807182266

Offset

0,3

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

B. Berselli, A description of the recursive method in Formula lines (first formula): website Matem@ticamente (in Italian).

Formula

a(n) = n*A123095(n) - Sum_{i=0..n-1} A123095(i). - Bruno Berselli, Apr 27 2010

a(n) = n * (n+1) * (2*n+1) * (105*n^10 +525*n^9 +525*n^8 -1050*n^7 -1190*n^6 +2310*n^5 +1420*n^4 -3285*n^3 -287*n^2 +2073*n -691)/2730. - Bruno Berselli, Oct 03 2010

a(n) = (-1)*Sum_{j=1..12} j*s(n+1,n+1-j)*S(n+12-j,n), where s(n,k) and S(n,k) are the Stirling numbers of the first kind and the second kind, respectively. - Mircea Merca, Jan 25 2014

Maple

[seq(add(i^12, i=1..n), n=0..18)];

Mathematica

Table[Sum[k^12, {k, n}], {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Range[0, 30]^12]  (* Harvey P. Dale, Apr 26 2011 *)

Prog

(Sage) [bernoulli_polynomial(n, 13)/13 for n in range(1, 30)] # Zerinvary Lajos, May 17 2009
(Python)
A123094_list, m = [0], [479001600, -2634508800, 6187104000, -8083152000, 6411968640, -3162075840, 953029440, -165528000, 14676024, -519156, 4094, -1, 0 , 0]
for _ in range(10**2):
    for i in range(13):
        m[i+1]+= m[i]
    A123094_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014
(Magma) [(&+[j^12: j in [0..n]]): j in [0..30]]; // G. C. Greubel, Jul 21 2021

Crossrefs

Sequences of the form Sum_{j=0..n} j^m : A000217 (m=1), A000330 (m=2), A000537 (m=3), A000538 (m=4), A000539 (m=5), A000540 (m=6), A000541 (m=7), A000542 (m=8), A007487 (m=9), A023002 (m=10), A123095 (m=11), this sequence (m=12), A181134 (m=13).

Cf. A008456, A215083.

Cf. A008275, A008277.

Sequence in context: A017687 A013960 A036090 * A226695 A031562 A345507

Adjacent sequences: A123091 A123092 A123093 * A123095 A123096 A123097

Keyword

easy,nonn

Author

Zerinvary Lajos, Sep 27 2006

Status

approved