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A023002 Sum of 10th powers. 7
0, 1, 1025, 60074, 1108650, 10874275, 71340451, 353815700, 1427557524, 4914341925, 14914341925, 40851766526, 102769130750, 240627622599, 529882277575, 1106532668200, 2206044295976, 4222038196425, 7792505423049, 13923571680850 (list; graph; refs; listen; history; text; internal format)
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 (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*s(n+1,n+1-j)*S(n+10-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

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

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Last modified August 5 07:37 EDT 2021. Contains 346464 sequences. (Running on oeis4.)