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).
1, -8, 1, 32, -3, -384, 30, 1, 1536, -105, -5, -30720, 1890, 105, 2, 61440, -3465, -210, -7, -10321920, 540540, 34650, 1512, 17, 4587520, -225225, -15015, -770, -17, -1486356480, 68918850, 4729725, 270270, 8415, 62, 2972712960, -130945815, -9189180, -567567, -21879, -341
Offset: 1
Examples
S(1) = ( 1 ) / (2), S(2) = ( -8 + Pi^2 ) / (2^4) = A123092, S(3) = ( 32 - 3*Pi^2 ) / (2^5 * 2!) = A248895, S(4) = ( -384 + 30*Pi^2 + Pi^4 ) / (2^7 * 3!) = A248896, S(5) = ( 1536 - 105*Pi^2 - 5*Pi^4 ) / (2^7 * 4!), S(6) = ( -30720 + 1890*Pi^2 + 105*Pi^4 + 2*Pi^6 ) / (2^9 * 5!), S(7) = ( 61440 - 3465*Pi^2 - 210*Pi^4 - 7*Pi^6 ) / (2^10 * 5!), S(8) = (-10321920 + 540540*Pi^2 + 34650*Pi^4 + 1512*Pi^6 + 17*Pi^8) / (2^12 * 7!), S(9) = ( 4587520 - 225225*Pi^2 - 15015*Pi^4 - 770*Pi^6 - 17*Pi^8) / (2^18 * 5 * 7), ...
References
- E. P. Adams, Smithsonian Mathematical Formulae and Tables of Elliptic Functions, 1922 (eq. 6.911).
Links
- I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.235).
- Sean A. Irvine, Computing A382782, 2025.
- Sean A. Irvine, Java program (github)
- L. B. W. Jolley, Summation of Series, Dover, (1961) (eq. 373).
Comments