|
%I #13 Sep 14 2013 04:49:09
%S 6,60,60,672,112,672,8160,544,544,8160,523776,130944,261888,130944,
%T 523776,1397760,93184,6656,6656,93184,1397760,3121152,22368256,
%U 44736512,11184128,44736512,22368256,3121152,268431360
%N Triangle read by rows, whose row sums using Euler numbers are the unsigned even-indexed Bernoulli numbers (denominators).
%D George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 132.
%F T(n, k) = denominator(-(-1)^n*n*binomial(2n-2, 2k)*E(2k)*E(2n-2k-2)/(2^(2n-1)*(2^(2n)-1))), where E(.) = Euler number.
%e 1/6;
%e 1/60, 1/60;
%e 5/672, 1/112, 5/672;
%e 61/8160, 5/544, 5/544, 61/8160;
%e 6925/523776, 2135/130944, 4375/261888, 2135/130944, 6925/523776;
%e ...
%e Row sums are 1/6, 1/30, 1/42, 1/30, 5/66, ...
%e From _Bruno Berselli_, Sep 14 2013: (Start)
%e Triangle begins:
%e 6;
%e 60, 60;
%e 672, 112, 672;
%e 8160, 544, 544, 8160;
%e 523776, 130944, 261888, 130944, 523776;
%e 1397760, 93184, 6656, 6656, 93184, 1397760;
%e 3121152, 22368256, 44736512, 11184128, 44736512, 22368256, 3121152, etc.
%e (End)
%t t[n_, k_] := -(-1)^n n Binomial[2 n - 2, 2 k] EulerE[2 k] EulerE[2 n - 2 k - 2]/(2^(2 n - 1) (2^(2 n) - 1)); Table[t[n, k], {n, 1, 8}, {k, 0, n - 1}] // Flatten // Denominator
%Y Cf. A229096 (numerators), A002445, A000364, A000367.
%K nonn,frac,tabl
%O 1,1
%A _Jean-François Alcover_, Sep 13 2013
|
