cp's OEIS Frontend

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)!.

%I A382784 #4 Apr 05 2025 17:26:41
%S A382784 2,-8,1,64,-6,-768,60,2,12288,-840,-40,-245760,15120,840,16,5898240,
%T A382784 -332640,-20160,-672,-165150720,8648640,554400,24192,272,5284823040,
%U A382784 -259459200,-17297280,-887040,-19584,-190253629440,8821612800,605404800,34594560,1077120,7936,7610145177600,-335221286400,-23524300800,-1452971520,-56010240,-872960
%N 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)!.
%C A382784 See A382782 for a version of this triangle where common factors have been removed.
%e A382784 Triangle begins:
%e A382784 S(1) =  (2) / (2^2 * 0!),
%e A382784 S(2) = -(8 - Pi^2) / (2^4 * 1!) = A123092,
%e A382784 S(3) =  (64 - 6*Pi^2) / (2^6 * 2!) = A248895,
%e A382784 S(4) = -(768 - 60*Pi^2 - 2*Pi^4)/ (2^8 * 3!) = A248896,
%e A382784 S(5) =  (12288 - 840*Pi^2 - 40*Pi^4) / (2^10 * 4!),
%e A382784 S(6) = -(245760 - 15120*Pi^2 - 840*Pi^4 - 16*Pi^6) / (2^12 * 5!),
%e A382784 S(7) =  (5898240 - 332640*Pi^2 - 20160*Pi^4 - 672*Pi^6) / (2^14 * 6!),
%e A382784 S(8) = -(165150720 - 8648640*Pi^2 - 554400*Pi^4 - 24192*Pi^6 - 272*Pi^8) / (2^16 * 7!),
%e A382784 S(9) =  (5284823040 - 259459200*Pi^2 - 17297280*Pi^4 - 887040*Pi^6 - 19584*Pi^8) / (2^18 * 8!), ...
%Y A382784 Cf. A123092 (n=2), A248895 (n=3), A248896 (n=4).
%Y A382784 Cf. A382782.
%K A382784 sign
%O A382784 1,1
%A A382784 _Sean A. Irvine_, Apr 04 2025