A382784 Irregular triangle T(n,k) read by rows of the coefficients of Pi^(2k) in the expansion of Sum_{k>=1} (1 / (4k^2-1)^n) with denominator 2^(2n)*(n-1)!.

2, -8, 1, 64, -6, -768, 60, 2, 12288, -840, -40, -245760, 15120, 840, 16, 5898240, -332640, -20160, -672, -165150720, 8648640, 554400, 24192, 272, 5284823040, -259459200, -17297280, -887040, -19584, -190253629440, 8821612800, 605404800, 34594560, 1077120, 7936, 7610145177600, -335221286400, -23524300800, -1452971520, -56010240, -872960

Offset: 1

Author

Sean A. Irvine, Apr 04 2025

Keywords

• sign

Comments

See A382782 for a version of this triangle where common factors have been removed.

Examples

			Triangle begins:
S(1) =  (2) / (2^2 * 0!),
S(2) = -(8 - Pi^2) / (2^4 * 1!) = A123092,
S(3) =  (64 - 6*Pi^2) / (2^6 * 2!) = A248895,
S(4) = -(768 - 60*Pi^2 - 2*Pi^4)/ (2^8 * 3!) = A248896,
S(5) =  (12288 - 840*Pi^2 - 40*Pi^4) / (2^10 * 4!),
S(6) = -(245760 - 15120*Pi^2 - 840*Pi^4 - 16*Pi^6) / (2^12 * 5!),
S(7) =  (5898240 - 332640*Pi^2 - 20160*Pi^4 - 672*Pi^6) / (2^14 * 6!),
S(8) = -(165150720 - 8648640*Pi^2 - 554400*Pi^4 - 24192*Pi^6 - 272*Pi^8) / (2^16 * 7!),
S(9) =  (5284823040 - 259459200*Pi^2 - 17297280*Pi^4 - 887040*Pi^6 - 19584*Pi^8) / (2^18 * 8!), ...
		

Crossrefs

Cf. A123092 (n=2), A248895 (n=3), A248896 (n=4).

Cf. A382782.