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A279396
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Triangle read by rows T(n, m) = sigma^*_(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^*_(k)(n) given in a comment in A279395.
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9
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1, 1, 0, 1, 1, 2, 1, 3, 4, 1, 1, 7, 10, 5, 2, 1, 15, 28, 19, 6, 0, 1, 31, 82, 71, 26, 4, 2, 1, 63, 244, 271, 126, 30, 8, 2, 1, 127, 730, 1055, 626, 196, 50, 13, 3, 1, 255, 2188, 4159, 3126, 1230, 344, 83, 13, 0, 1, 511, 6562, 16511, 15626, 7564, 2402, 583, 91, 6, 2, 1, 1023, 19684, 65791, 78126, 45990, 16808, 4367, 757, 78, 12, 2
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OFFSET
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1,6
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COMMENTS
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The array A(k, n) = sigma^*_(k)(n) (notation of the Hardy reference, given also in a comment in A279395) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^k, for k >= 0 and n >=1, has the rows A112329, A113184, A064027, A008457, A279395, for k=0..4.
The triangle T(n, m) is obtained from the array A(k, n) read by upwards antidiagonals, with offset n=1.
The diagonals of triangle T are the rows of the array A. Each diagonal is multiplicative. See the given A-numbers above.
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REFERENCES
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G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.
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LINKS
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FORMULA
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T(n, m) = Sum_{ d >= 1, d divides m} (-1)^(m-d)*d^(n-m) = sigma^*_(n-m)(m), n >= 1, m = 1,2, ..., n. For the definition of
sigma^*_(k)(n) see the Hardy reference or a comment in A279395.
O.g.f triangle T: G(z, x) = Sum_{m>=0}
G(m, z)*(x*z)^m, with the column o.g.f. G( m, z) (with offset 0) given in a comment above.
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EXAMPLE
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The triangle T(n, m) begins:
n\m 1 2 3 4 5 6 7 8 9 10
1: 1
2: 1 0
3: 1 1 2
4: 1 3 4 1
5: 1 7 10 5 2
6: 1 15 28 19 6 0
7: 1 31 82 71 26 4 2
8: 1 63 244 271 126 30 8 2
9: 1 127 730 1055 626 196 50 13 3
10: 1 255 2188 4159 3126 1230 344 83 13 0
...
n = 11: 1 511 6562 16511 15626 7564 2402 583 91 6 2,
n = 12: 1 1023 19684 65791 78126 45990 16808 4367 757 78 12 2.
n = 13: 1 2047 59050 262655 390626 277876 117650 33823 6643 882 122 20 2,
n = 14: 1 4095 177148 1049599 1953126 1673310 823544 266303 59293 9390 1332 190 14 0,
n = 15: 1 8191 531442 4196351 9765626 10058524 5764802 2113663 532171 96906 14642 1988 170 8 4.
...
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CROSSREFS
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Cf. A279394, A112329, A113184, A064027, A008457, A279395, A000012, A000225, A034472, A099393, A034474.
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KEYWORD
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AUTHOR
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STATUS
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approved
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