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A229096 Triangle read by rows, whose row sums using Euler numbers are the unsigned even-indexed Bernoulli numbers (numerators). 1

%I

%S 1,1,1,5,1,5,61,5,5,61,6925,2135,4375,2135,6925,50521,4155,305,305,

%T 4155,50521,439985,3890117,7998375,2005619,7998375,3890117,439985,

%U 199360981,49190323,50571521,16913897,16913897,50571521,49190323,199360981

%N Triangle read by rows, whose row sums using Euler numbers are the unsigned even-indexed Bernoulli numbers (numerators).

%D George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 132.

%F T(n, k) = numerator(-(-1)^n*n*binomial(2n-2, 2k)*E(2k)*E(2n-2k-2)/(2^(2n-1)*(2^(2n)-1))), where with 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 1;

%e 1, 1;

%e 5, 1, 5;

%e 61, 5, 5, 61;

%e 6925, 2135, 4375, 2135, 6925;

%e 50521, 4155, 305, 305, 4155, 50521, etc. (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 // Numerator

%Y Cf. A229097(denominators), A002445, A000364, A000367.

%K nonn,frac,tabl

%O 1,4

%A _Jean-Fran├žois Alcover_, Sep 13 2013

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Last modified September 15 20:35 EDT 2019. Contains 327087 sequences. (Running on oeis4.)