A023002 Sum of 10th powers.

0, 1, 1025, 60074, 1108650, 10874275, 71340451, 353815700, 1427557524, 4914341925, 14914341925, 40851766526, 102769130750, 240627622599, 529882277575, 1106532668200, 2206044295976, 4222038196425, 7792505423049, 13923571680850

Offset

0,3

Links

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

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

Eric Weisstein's World of Mathematics, Power Sum.

Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Formula

a(n) = n*(n+1)*(2*n+1)*(n^2+n-1)(3*n^6 +9*n^5 +2*n^4 -11*n^3 +3*n^2 +10*n -5)/66 (see MathWorld, Power Sum, formula 40). - Bruno Berselli, Apr 26 2010

a(n) = n*A007487(n) - Sum_{i=0..n-1} A007487(i). - Bruno Berselli, Apr 27 2010

From Bruno Berselli, Aug 23 2011: (Start)

a(n) = -a(-n-1).

G.f.: x*(1+x)*(1 +1012*x +46828*x^2 +408364*x^3 +901990*x^4 +408364*x^5 +46828*x^6 +1012*x^7 +x^8)/(1-x)^12. (End)

a(n) = (-1)*Sum_{j=1..10} j*Stirling1(n+1,n+1-j)*Stirling2(n+10-j,n). - Mircea Merca, Jan 25 2014

Maple

A023002:= n-> bernoulli(11, n+1)/11; seq(A023002(n), n=0..30); # G. C. Greubel, Jul 21 2021

Mathematica

Table[Sum[k^10, {k, n}], {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Range[0, 20]^10] (* Harvey P. Dale, Aug 23 2011 *)

Prog

(Sage) [bernoulli_polynomial(n, 11)/11 for n in range(2, 21)]# Zerinvary Lajos, May 17 2009
(Magma) [&+[n^10: n in [0..m]]: m in [0..19]];  // Bruno Berselli, Aug 23 2011
(PARI) a(n)=(6*x^11+33*x^10+55*x^9-66*x^7+66*x^5-33*x^3+5*x)/66 \\ Charles R Greathouse IV, Aug 23 2011
(PARI) a(n)=sum(i=0, 10, binomial(11, i)*bernfrac(i)*n^(11-i))/11+n^10 \\ Charles R Greathouse IV, Aug 23 2011
(Python)
A023002_list, m = [0], [3628800, -16329600, 30240000, -29635200, 16435440, -5103000, 818520, -55980, 1022, -1, 0 , 0]
for _ in range(20):
    for i in range(11):
        m[i+1]+= m[i]
    A023002_list.append(m[-1])
print(A023002_list) # Chai Wah Wu, Nov 05 2014

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), this sequence (m=10), A123095 (m=11), A123094 (m=12), A181134 (m=13).

Row 10 of array A103438.

Cf. A215083.

Sequence in context: A013958 A294305 A036088 * A279643 A168119 A272672

Adjacent sequences: A022999 A023000 A023001 * A023003 A023004 A023005

Keyword

nonn,easy

Author

David W. Wilson

Status

approved