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A382782 Irregular triangle T(n,k) read by rows of the reduced coefficients of Pi^(2*k) in the expansion of Sum_{k>=1} (1 / (4*k^2-1)^n).

%I A382782 #14 Apr 14 2025 07:39:01
%S A382782 1,-8,1,32,-3,-384,30,1,1536,-105,-5,-30720,1890,105,2,61440,-3465,
%T A382782 -210,-7,-10321920,540540,34650,1512,17,4587520,-225225,-15015,-770,
%U A382782 -17,-1486356480,68918850,4729725,270270,8415,62,2972712960,-130945815,-9189180,-567567,-21879,-341
%N A382782 Irregular triangle T(n,k) read by rows of the reduced coefficients of Pi^(2*k) in the expansion of Sum_{k>=1} (1 / (4*k^2-1)^n).
%C A382782 The expansion of S(n) = Sum_{k>=1} (1 / (4*k^2-1)^n) in even powers of Pi was apparently first found by Euler and the solution for n<=4 appear in many tables of sums.
%C A382782 These sums have a natural denominator of 2^(2*n)*(n-1)! (or, more precisely, 2^(2*n+floor((n-1)/2))*(n-1)!), but sometimes (e.g., n=7, n=9) there are additional common factors leading to the "reduced" triangle presented here.
%D A382782 E. P. Adams, Smithsonian Mathematical Formulae and Tables of Elliptic Functions, 1922 (eq. 6.911).
%H A382782 I. S. Gradsteyn and I. M. Ryzhik, <a href="http://mathtable.com/gr/index.html">Table of integrals, series and products</a> (6th ed.), 2000, (eq. 0.235).
%H A382782 Sean A. Irvine, <a href="/A382782/a382782.pdf">Computing A382782</a>, 2025.
%H A382782 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a382/A382782.java">Java program</a> (github)
%H A382782 L. B. W. Jolley, <a href="https://archive.org/details/summationofserie00joll">Summation of Series</a>, Dover, (1961) (eq. 373).
%e A382782 S(1) = (        1                                                 ) / (2),
%e A382782 S(2) = (       -8 +        Pi^2                                   ) / (2^4) = A123092,
%e A382782 S(3) = (       32 -      3*Pi^2                                   ) / (2^5 * 2!) = A248895,
%e A382782 S(4) = (     -384 +     30*Pi^2 +       Pi^4                      ) / (2^7 * 3!) = A248896,
%e A382782 S(5) = (     1536 -    105*Pi^2 -     5*Pi^4                      ) / (2^7 * 4!),
%e A382782 S(6) = (   -30720 +   1890*Pi^2 +   105*Pi^4 +    2*Pi^6          ) / (2^9 * 5!),
%e A382782 S(7) = (    61440 -   3465*Pi^2 -   210*Pi^4 -    7*Pi^6          ) / (2^10 * 5!),
%e A382782 S(8) = (-10321920 + 540540*Pi^2 + 34650*Pi^4 + 1512*Pi^6 + 17*Pi^8) / (2^12 * 7!),
%e A382782 S(9) = (  4587520 - 225225*Pi^2 - 15015*Pi^4 -  770*Pi^6 - 17*Pi^8) / (2^18 * 5 * 7), ...
%Y A382782 Cf. A123092 (n=2), A248895 (n=3), A248896 (n=4).
%Y A382782 Cf. A382783, A382784.
%K A382782 sign,tabf
%O A382782 1,2
%A A382782 _Sean A. Irvine_, Apr 04 2025