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Search: id:A103438
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| A103438 |
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Square array T(m,n) read by antidiagonals: Sum[k=0..n, k^m ]. |
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24
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| 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; listen)
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OFFSET
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0,6
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REFERENCES
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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.
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LINKS
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V. J. W. Guo and J. Zeng, A q-analogue of Faulhaber's formula for sums of powers
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
D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 61 (1993), no. 203, 277-294.
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FORMULA
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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. [From Peter Luschny (peter(AT)luschny.de), Nov 16 2008]
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EXAMPLE
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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,
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MAPLE
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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]
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CROSSREFS
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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.
Adjacent sequences: A103435 A103436 A103437 this_sequence A103439 A103440 A103441
Sequence in context: A155584 A139600 A166278 this_sequence A068920 A099390 A124031
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KEYWORD
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nonn,tabl
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AUTHOR
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Ralf Stephan, Feb 11 2005
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