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A229096
Triangle read by rows, whose row sums using Euler numbers are the unsigned even-indexed Bernoulli numbers (numerators).
1
1, 1, 1, 5, 1, 5, 61, 5, 5, 61, 6925, 2135, 4375, 2135, 6925, 50521, 4155, 305, 305, 4155, 50521, 439985, 3890117, 7998375, 2005619, 7998375, 3890117, 439985, 199360981, 49190323, 50571521, 16913897, 16913897, 50571521, 49190323, 199360981
OFFSET
1,4
REFERENCES
George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 132.
FORMULA
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.
EXAMPLE
1/6;
1/60, 1/60;
5/672, 1/112, 5/672;
61/8160, 5/544, 5/544, 61/8160;
6925/523776, 2135/130944, 4375/261888, 2135/130944, 6925/523776;
...
Row sums are 1/6, 1/30, 1/42, 1/30, 5/66, ...
From Bruno Berselli, Sep 14 2013: (Start)
Triangle begins:
1;
1, 1;
5, 1, 5;
61, 5, 5, 61;
6925, 2135, 4375, 2135, 6925;
50521, 4155, 305, 305, 4155, 50521, etc. (End)
MATHEMATICA
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
CROSSREFS
Cf. A229097(denominators), A002445, A000364, A000367.
Sequence in context: A281105 A327283 A105060 * A290797 A375066 A200423
KEYWORD
nonn,frac,tabl
AUTHOR
STATUS
approved