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 A103438 Square array T(m,n) read by antidiagonals: Sum_{k=1..n} k^m. 40
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified April 24 17:29 EDT 2024. Contains 371962 sequences. (Running on oeis4.)