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

0,6

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

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

Crossrefs

Rows include A000027, A000217, A000330, A000537, A000538, A000539, A000540, A000541, A000542, A007487, A023002.

Columns include A000051, A001550, A001551, A001552, A001553, A001554, A001555, A001556, A001557.

Diagonals include A076015 and A031971.

Antidiagonal sums are in A103439.

Antidiagonals are the rows of triangle A192001.

Cf. A065551, A093556.

Sequence in context: A365515 A316269 A242379 * A291556 A323073 A167279

Adjacent sequences: A103435 A103436 A103437 * A103439 A103440 A103441

Keyword

nonn,tabl

Author

Ralf Stephan, Feb 11 2005

Status

approved