cp's OEIS Frontend

A193354 Triangle read by rows: T(n,k) = (-1)^(n-k) * r16(n-k) * 2^(3*b(k)) * sigma_3(O(k)), for k=1 to n, for n>=1 (see comments for terms used).

%I A193354 #21 Jan 07 2025 01:58:52
%S A193354 1,-32,8,480,-256,28,-4480,3840,-896,64,29152,-35840,13440,-2048,126,
%T A193354 -140736,233216,-125440,30720,-4032,224,525952,-1125888,816256,
%U A193354 -286720,60480,-7168,344,-1580800,4207616,-3940608,1865728,-564480,107520,-11008,512
%N A193354 Triangle read by rows: T(n,k) = (-1)^(n-k) * r16(n-k) * 2^(3*b(k)) * sigma_3(O(k)), for k=1 to n, for n>=1 (see comments for terms used).
%C A193354 Functions used in name: r16(n) is A000152(n), O(n) is A000265(n), b(n) is A007814(n).
%H A193354 Amiram Eldar, <a href="/A193354/b193354.txt">Table of n, a(n) for n = 1..5050</a> (first 100 rows)
%H A193354 John A. Ewell, <a href="http://dx.doi.org/10.1090/S0002-9939-1984-0735559-9">A formula for Ramanujan's tau function</a>, Proc. Amer. Math. Soc. 91 (1984), 37-40.
%F A193354 For n>=1, Sum_{k=1, n} a(k) = A000594(n).
%e A193354 Triangle starts:
%e A193354   1: 1
%e A193354   2: -32, 8
%e A193354   3: 480, -256, 28
%e A193354   4: -4480, 3840, -896, 64
%e A193354   5: 29152, -35840, 13440, -2048, 126
%t A193354 T[n_, k_] := Module[{e = IntegerExponent[k, 2]}, (-1)^(n - k) * SquaresR[16, n - k]*2^(3*e)*DivisorSigma[3, k/2^e]]; Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Jan 06 2025 *)
%Y A193354 Cf. A000152, A000265, A000594, A007814, A001158, A051000.
%K A193354 sign,tabl
%O A193354 1,2
%A A193354 _Michel Marcus_, Dec 20 2012