OFFSET
0,6
COMMENTS
For the o.g.f.s of the column sequences for this array, see A196837 and the link given there. - Wolfdieter Lang, Oct 15 2011
T(m,n)/n is the m-th moment of the discrete uniform distribution on {1,2,...,n}. - Geoffrey Critzer, Dec 31 2018
T(1,n) divides T(m,n) for odd m. - Franz Vrabec, Dec 23 2020
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
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
EXAMPLE
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;
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
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 *)
PROG
(PARI) T(m, n)=sum(k=0, n, k^m)
(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
(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
(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)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Ralf Stephan, Feb 11 2005
STATUS
approved