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).

1, -32, 8, 480, -256, 28, -4480, 3840, -896, 64, 29152, -35840, 13440, -2048, 126, -140736, 233216, -125440, 30720, -4032, 224, 525952, -1125888, 816256, -286720, 60480, -7168, 344, -1580800, 4207616, -3940608, 1865728, -564480, 107520, -11008, 512

Offset: 1

Author

Michel Marcus, Dec 20 2012

Keywords

• sign
• tabl

Comments

Functions used in name: r16(n) is A000152(n), O(n) is A000265(n), b(n) is A007814(n).

Examples

			Triangle starts:
  1: 1
  2: -32, 8
  3: 480, -256, 28
  4: -4480, 3840, -896, 64
  5: 29152, -35840, 13440, -2048, 126
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..5050 (first 100 rows)
• John A. Ewell, A formula for Ramanujan's tau function, Proc. Amer. Math. Soc. 91 (1984), 37-40.

Crossrefs

Cf. A000152, A000265, A000594, A007814, A001158, A051000.

Programs

Mathematica

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 *)

Formula

For n>=1, Sum_{k=1, n} a(k) = A000594(n).