A103438 Square array T(m,n) read by antidiagonals: Sum_{k=1..n} k^m.
0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 5, 6, 4, 0, 1, 9, 14, 10, 5, 0, 1, 17, 36, 30, 15, 6, 0, 1, 33, 98, 100, 55, 21, 7, 0, 1, 65, 276, 354, 225, 91, 28, 8, 0, 1, 129, 794, 1300, 979, 441, 140, 36, 9, 0, 1, 257, 2316, 4890, 4425, 2275, 784, 204, 45, 10
Offset: 0
Examples
Square array begins: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ... A001477; 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ... A000217; 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ... A000330; 0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, ... A000537; 0, 1, 17, 98, 354, 979, 2275, 4676, 8772, 15333, ... A000538; 0, 1, 33, 276, 1300, 4425, 12201, 29008, 61776, 120825, ... A000539; 0, 1, 65, 794, 4890, 20515, 67171, 184820, 446964, 978405, ... A000540; Antidiagonal triangle begins as: 0; 0, 1; 0, 1, 2; 0, 1, 3, 3; 0, 1, 5, 6, 4; 0, 1, 9, 14, 10, 5; 0, 1, 17, 36, 30, 15, 6;
References
- J. Faulhaber, Academia Algebrae, Darinnen die miraculosische inventiones zu den höchsten Cossen weiters continuirt und profitirt werden, Augspurg, bey Johann Ulrich Schönigs, 1631.
Links
- G. C. Greubel, Antidiagonals n = 0..50, flattened
- José L. Cereceda, Sums of powers of integers and hyperharmonic numbers, arXiv:2005.03407 [math.NT], 2020.
- T. A. Gulliver, Divisibility of sums of powers of odd integers, Int. Math. For. 5 (2010) 3059-3066.
- T. A. Gulliver, Sums of Powers of Integers Divisible by Three, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 38, pp. 1895-1901. - From _N. J. A. Sloane_, Dec 22 2012
- V. J. W. Guo and J. Zeng, A q-analogue of Faulhaber's formula for sums of powers, arXiv:math/0501441 [math.CO], 2005.
- H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
- T. Kim, q-analogues of the sums of powers of consecutive integers, arXiv:math/0502113 [math.NT], 2005.
- D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 61 (1993), no. 203, 277-294.
- Eric Weisstein's World of Mathematics, Discrete Uniform Distribution.
- Wikipedia, Faulhaber's formula
Crossrefs
Programs
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Magma
T:= func< n,k | n eq 0 select k else (&+[j^n: j in [0..k]]) >; [T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 22 2021
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Maple
seq(print(seq(Zeta(0,-k,1)-Zeta(0,-k,n+1),n=0..9)),k=0..6); # (Produces the square array from the example.) Peter Luschny, Nov 16 2008 # alternative A103438 := proc(m,n) (bernoulli(m+1,n+1)-bernoulli(m+1))/(m+1) ; if m = 0 then %-1 ; else % ; end if; end proc: # R. J. Mathar, May 10 2013 # simpler: A103438 := proc(m,n) (bernoulli(m+1,n+1)-bernoulli(m+1,1))/(m+1) ; end proc: # Peter Luschny, Mar 20 2024
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Mathematica
T[m_, n_]:= HarmonicNumber[m, -n]; Flatten[Table[T[m-n, n], {m, 0, 11}, {n, m, 0, -1}]] (* Jean-François Alcover, May 11 2012 *) -
PARI
T(m,n)=sum(k=0,n,k^m)
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Python
from itertools import count, islice from math import comb from fractions import Fraction from sympy import bernoulli def A103438_T(m,n): return sum(k**m for k in range(1,n+1)) if n<=m else int(sum(comb(m+1,i)*(bernoulli(i) if i!=1 else Fraction(1,2))*n**(m-i+1) for i in range(m+1))/(m+1)) def A103438_gen(): # generator of terms for m in count(0): for n in range(m+1): yield A103438_T(m-n,n) A103438_list = list(islice(A103438_gen(),100)) # Chai Wah Wu, Oct 23 2024
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SageMath
def T(n,k): return (bernoulli_polynomial(k+1, n+1) - bernoulli_polynomial(1, n+1)) /(n+1) flatten([[T(n-k,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 22 2021
Formula
E.g.f.: e^x*(e^(x*y)-1)/(e^x-1).
T(m, n) = Zeta(-n, 1) - Zeta(-n, m + 1), for m >= 0 and n >= 0, where Zeta(z, v) is the Hurwitz zeta function. - Peter Luschny, Nov 16 2008
T(m, n) = HarmonicNumber(m, -n). - Jean-François Alcover, May 11 2012
T(m, n) = (Bernoulli(m + 1, n + 1) - Bernoulli(m + 1, 1)) / (m + 1). - Peter Luschny, Mar 20 2024
T(m, n) = Sum_{k=0...m-n} B(k)*(-1)^k*binomial(m-n,k)*n^(m-n-k+1)/(m-n-k+1), where B(k) = Bernoulli number A027641(k) / A027642(k). - Robert B Fowler, Aug 20 2024
T(m, n) = Sum_{i=1..n} J_m(i)*floor(n/i), where J_m is the m-th Jordan totient function. - Ridouane Oudra, Jul 19 2025
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