A279396 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.

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

Offset: 1

Author

Wolfdieter Lang, Jan 10 2017

Keywords

• nonn
• tabl
• easy

Comments

The array A(k, n) = sigma^*A279395)%20=%20Sum">(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.

The row sums are given in A279397.

The column sums (with offset 0) are A000012, A000225, A034472, A099393, A034474, .. with o.g.f. G(m, z) = (-1)^m*Sum_{d | m} (-1)^d/(1 - d*z), m >= 1.

Examples

			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.
...
		

References

• G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Crossrefs

Cf. A279394, A112329, A113184, A064027, A008457, A279395, A000012, A000225, A034472, A099393, A034474.

Formula

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.