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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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.

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=1..78.

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.

EXAMPLE

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.

...

CROSSREFS

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

Sequence in context: A181642 A011358 A160188 * A161224 A147567 A247045

Adjacent sequences:  A279393 A279394 A279395 * A279397 A279398 A279399

KEYWORD

nonn,tabl,easy

AUTHOR

Wolfdieter Lang, Jan 10 2017

STATUS

approved

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Last modified July 27 04:25 EDT 2017. Contains 289841 sequences.