A023002
Sum of 10th powers.
Original entry on oeis.org
0, 1, 1025, 60074, 1108650, 10874275, 71340451, 353815700, 1427557524, 4914341925, 14914341925, 40851766526, 102769130750, 240627622599, 529882277575, 1106532668200, 2206044295976, 4222038196425, 7792505423049, 13923571680850
Offset: 0
- 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).
- Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
- 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).
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).
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[&+[n^10: n in [0..m]]: m in [0..19]]; // Bruno Berselli, Aug 23 2011
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A023002:= n-> bernoulli(11, n+1)/11; seq(A023002(n), n=0..30); # G. C. Greubel, Jul 21 2021
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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 *)
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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
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a(n)=sum(i=0,10,binomial(11,i)*bernfrac(i)*n^(11-i))/11+n^10 \\ Charles R Greathouse IV, Aug 23 2011
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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
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[bernoulli_polynomial(n,11)/11 for n in range(2, 21)]# Zerinvary Lajos, May 17 2009
A123094
Sum of first n 12th powers.
Original entry on oeis.org
0, 1, 4097, 535538, 17312754, 261453379, 2438235715, 16279522916, 84998999652, 367428536133, 1367428536133, 4505856912854, 13421957361110, 36720042483591, 93413954858887, 223160292749512, 504635269460168, 1087257506689929, 2244088888116105, 4457403807182266
Offset: 0
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).
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[(&+[j^12: j in [0..n]]): j in [0..30]]; // G. C. Greubel, Jul 21 2021
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[seq(add(i^12, i=1..n), n=0..18)];
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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 *)
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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
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[bernoulli_polynomial(n,13)/13 for n in range(1, 30)] # Zerinvary Lajos, May 17 2009
A123095
Sum of first n 11th powers.
Original entry on oeis.org
0, 1, 2049, 179196, 4373500, 53201625, 415998681, 2393325424, 10983260016, 42364319625, 142364319625, 427675990236, 1170684360924, 2962844754961, 7012409924625, 15662165784000, 33254351828416, 67526248136049, 131794658215281, 248284917113500, 453084917113500
Offset: 0
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
- Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
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), this sequence (m=11),
A123094 (m=12),
A181134 (m=13).
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[(&+[j^11: j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jul 21 2021
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[seq(add(i^11, i=1..n), n=0..20)];
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^11 od: seq(a[n], n=0..13); # Zerinvary Lajos, Feb 22 2008
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Table[Sum[k^11, {k, n}], {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Range[0,20]^11] (* Harvey P. Dale, Sep 17 2021 *)
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A123095_list, m = [0], [39916800, -199584000, 419126400, -479001600, 322494480, -129230640, 29607600, -3498000, 171006, -2046, 1, 0 , 0]
for _ in range(10**2):
for i in range(12):
m[i+1]+= m[i]
A123095_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014
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[(bernoulli_polynomial(n+1, 12) - bernoulli(12))/12 for n in (0..30)] # G. C. Greubel, Jul 21 2021
Showing 1-3 of 3 results.