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A279396 Triangle read by rows T(n, m) = sigma^*(n-m)(m),</span> n <span class="maths">>= 1,</span> m <span class="maths">= 1, 2,</span> ..., n, with <span class="maths">sigma^*(k)(n) given in a comment in A279395.

%I A279396 #7 Jan 10 2017 15:51:59
%S A279396 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 A279396 244,271,126,30,8,2,1,127,730,1055,626,196,50,13,3,1,255,2188,4159,
%U A279396 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 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.
%C A279396 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 A279396 The triangle T(n, m) is obtained from the array A(k, n) read by upwards antidiagonals, with offset n=1.
%C A279396 The diagonals of triangle T are the rows of the array A. Each diagonal is multiplicative. See the given A-numbers above.
%C A279396 The row sums are given in A279397.
%C A279396 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 A279396 G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.
%F A279396 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 A279396 sigma^*_(k)(n) see the Hardy reference or a comment in A279395.
%F A279396 O.g.f triangle T: G(z, x) = Sum_{m>=0}
%F A279396 G(m, z)*(x*z)^m, with the column o.g.f. G( m, z) (with offset 0) given in a comment above.
%e A279396 The triangle T(n, m) begins:
%e A279396 n\m 1   2    3    4    5    6   7  8  9 10
%e A279396 1:  1
%e A279396 2:  1   0
%e A279396 3:  1   1    2
%e A279396 4:  1   3    4    1
%e A279396 5:  1   7   10    5    2
%e A279396 6:  1  15   28   19    6    0
%e A279396 7:  1  31   82   71   26    4   2
%e A279396 8:  1  63  244  271  126   30   8  2
%e A279396 9:  1 127  730 1055  626  196  50 13  3
%e A279396 10: 1 255 2188 4159 3126 1230 344 83 13  0
%e A279396 ...
%e A279396 n = 11: 1 511 6562 16511 15626 7564 2402 583 91 6 2,
%e A279396 n = 12: 1 1023 19684 65791 78126 45990 16808 4367 757 78 12 2.
%e A279396 n = 13: 1 2047 59050 262655 390626 277876 117650 33823 6643 882 122 20 2,
%e A279396 n = 14: 1 4095 177148 1049599 1953126 1673310 823544 266303 59293 9390 1332 190 14 0,
%e A279396 n = 15: 1 8191 531442 4196351 9765626 10058524 5764802 2113663 532171 96906 14642 1988 170 8 4.
%e A279396 ...
%Y A279396 Cf. A279394, A112329, A113184, A064027, A008457, A279395, A000012, A000225, A034472, A099393, A034474.
%K A279396 nonn,tabl,easy
%O A279396 1,6
%A A279396 _Wolfdieter Lang_, Jan 10 2017