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A103438 Square array T(m,n) read by antidiagonals: Sum_{k=0..n} k^m. 30
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, 0, 1, 513, 6818 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

For the o.g.f.s for the column sequences of this array see A196837 and the link given there. - Wolfdieter Lang, Oct 15 2011

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

Table of n, a(n) for n=0..69.

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.CO/0501441

H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects

T. Kim, q-analogues of the sums of powers of consecutive integers, arXiv:math.CO/0502113

D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 61 (1993), no. 203, 277-294.

Wikipedia, Faulhaber's formula

FORMULA

E.g.f.: e^x*(e^(xy)-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

EXAMPLE

Square array begins:

0, 1,  2,   3,    4,     5,     6,      7,      8,      9, ...

0, 1,  3,   6,   10,    15,    21,     28,     36,     45, ...

0, 1,  5,  14,   30,    55,    91,    140,    204,    285, ...

0, 1,  9,  36,  100,   225,   441,    784,   1296,   2025, ...

0, 1, 17,  98,  354,   979,  2275,   4676,   8772,  15333, ...

0, 1, 33, 276, 1300,  4425, 12201,  29008,  61776, 120825, ...

0, 1, 65, 794, 4890, 20515, 67171, 184820, 446964, 978405, ...

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

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)

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.

Cf. A065551, A093556.

Sequence in context: A198321 A166278 A242379 * A167279 A068920 A099390

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 May 28 21:38 EDT 2017. Contains 287241 sequences.