Search: id:A103438
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%I A103438
%S A103438 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,
%T A103438 98,100,55,21,7,0,1,65,276,354,225,91,28,8,0,1,129,794,1300,979,441,140,
%U A103438 36,9,0,1,257,2316,4890,4425,2275,784,204,45,10,0,1,513,6818
%N A103438 Square array T(m,n) read by antidiagonals: Sum[k=0..n, k^m ].
%D A103438 J. Faulhaber, Academia Algebrae, Darinnen die miraculosische inventiones 
               zu den h\"ochsten Cossen weiters continuirt und profitirt werden, 
               Augspurg, bey Johann Ulrich Sch\"onigs, 1631.
%H A103438 V. J. W. Guo and J. Zeng, <a href="http://arXiv.org/abs/math.CO/0501441">
               A q-analogue of Faulhaber's formula for sums of powers</a>
%H A103438 H. Helfgott and I. M. Gessel, <a href="http://arXiv.org/abs/math.CO/9810143">
               Enumeration of tilings of diamonds and hexagons with defects</a>
%H A103438 T. Kim, <a href="http://arXiv.org/abs/math.CO/0502113">q-analogues of 
               the sums of powers of consecutive integers</a>
%H A103438 D. E. Knuth, <a href="http://www-cs-faculty.Stanford.EDU/~knuth/preprints.html">
               Johann Faulhaber and sums of powers</a>, Math. Comp. 61 (1993), no. 
               203, 277-294.
%F A103438 E.g.f.: e^x*(e^(xy)-1)/(e^x-1).
%F A103438 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. [From Peter Luschny (peter(AT)luschny.de), 
               Nov 16 2008]
%e A103438 0,1,2,3,4,5,6,7,8,9,
%e A103438 0,1,3,6,10,15,21,28,36,45,
%e A103438 0,1,5,14,30,55,91,140,204,285,
%e A103438 0,1,9,36,100,225,441,784,1296,2025,
%e A103438 0,1,17,98,354,979,2275,4676,8772,15333,
%e A103438 0,1,33,276,1300,4425,12201,29008,61776,120825,
%e A103438 0,1,65,794,4890,20515,67171,184820,446964,978405,
%p A103438 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. [From Peter Luschny (peter(AT)luschny.de), 
               Nov 16 2008]
%Y A103438 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.
%Y A103438 Cf. A065551, A093556.
%Y A103438 Sequence in context: A155584 A139600 A166278 this_sequence A167279 A068920 
               A099390
%Y A103438 Adjacent sequences: A103435 A103436 A103437 this_sequence A103439 A103440 
               A103441
%K A103438 nonn,tabl
%O A103438 0,6
%A A103438 Ralf Stephan, Feb 11 2005

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