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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. 9

%I

%S 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,

%T 244,271,126,30,8,2,1,127,730,1055,626,196,50,13,3,1,255,2188,4159,

%U 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

%N 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.

%C 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.

%C The triangle T(n, m) is obtained from the array A(k, n) read by upwards antidiagonals, with offset n=1.

%C The diagonals of triangle T are the rows of the array A. Each diagonal is multiplicative. See the given A-numbers above.

%C The row sums are given in A279397.

%C 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.

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

%F 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

%F sigma^*_(k)(n) see the Hardy reference or a comment in A279395.

%F O.g.f triangle T: G(z, x) = Sum_{m>=0}

%F G(m, z)*(x*z)^m, with the column o.g.f. G( m, z) (with offset 0) given in a comment above.

%e The triangle T(n, m) begins:

%e n\m 1 2 3 4 5 6 7 8 9 10

%e 1: 1

%e 2: 1 0

%e 3: 1 1 2

%e 4: 1 3 4 1

%e 5: 1 7 10 5 2

%e 6: 1 15 28 19 6 0

%e 7: 1 31 82 71 26 4 2

%e 8: 1 63 244 271 126 30 8 2

%e 9: 1 127 730 1055 626 196 50 13 3

%e 10: 1 255 2188 4159 3126 1230 344 83 13 0

%e ...

%e n = 11: 1 511 6562 16511 15626 7564 2402 583 91 6 2,

%e n = 12: 1 1023 19684 65791 78126 45990 16808 4367 757 78 12 2.

%e n = 13: 1 2047 59050 262655 390626 277876 117650 33823 6643 882 122 20 2,

%e n = 14: 1 4095 177148 1049599 1953126 1673310 823544 266303 59293 9390 1332 190 14 0,

%e n = 15: 1 8191 531442 4196351 9765626 10058524 5764802 2113663 532171 96906 14642 1988 170 8 4.

%e ...

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

%K nonn,tabl,easy

%O 1,6

%A _Wolfdieter Lang_, Jan 10 2017

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Last modified May 24 06:53 EDT 2019. Contains 323529 sequences. (Running on oeis4.)