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A356547
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Triangle read by rows. T(n, k) are the coefficients of polynomials p_n(x) based on the Eulerian numbers of second order representing the Bernoulli numbers as B_n = p_n(1) / (2*(2*n - 1)!).
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3
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1, 1, 0, 6, -4, 0, 120, -192, 72, 0, 5040, -15840, 13920, -3456, 0, 362880, -2096640, 3306240, -1918080, 345600, 0, 39916800, -413683200, 1053803520, -1064448000, 448519680, -62208000, 0, 6227020800, -114960384000, 447866496000, -699342336000, 506348236800, -164428185600, 18289152000, 0
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OFFSET
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0,4
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COMMENTS
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The Bernoulli numbers with B(1) = 1/2 can be represented as the weighted sum of Eulerian numbers of second order, where we use the definition as given by Graham et al., Eulerian2(n, k) = A201637(n, k). For n >= 1 we have
B_(n) = (1/2)*Sum_{k=0..n} (-1)^k*Eulerian2(n, k) / binomial(2*n - 1, k).
Although this representation looks classical it was apparently first proved by Majer in 2010; later Fu and recently O'Sullivan gave an alternative proof (see links).
An analogous representation based on the Eulerian numbers of first order is given in A356545.
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270. (Since the thirty-fourth printing, Jan. 2022, with B(1) = 1/2.)
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LINKS
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FORMULA
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Let p_n(x) = Sum_{k=0..n} Eulerian2(n, k)*k!*(2*n - k - 1)! * (-x)^k.
T(n, k) = [x^k] p_n(x).
T(n, k) = (-1)^k*Eulerian2(n, k)*k!*(2*n - k - 1)!.
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EXAMPLE
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The triangle T(n, k) of the coefficients, sorted in ascending order, starts:
[0] 1;
[1] 1, 0;
[2] 6, -4, 0;
[3] 120, -192, 72, 0;
[4] 5040, -15840, 13920, -3456, 0;
[5] 362880, -2096640, 3306240, -1918080, 345600, 0;
[6] 39916800, -413683200, 1053803520, -1064448000, 448519680, -62208000, 0;
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MAPLE
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E2 := proc(n, k) combinat:-eulerian2(n, k) end:
p := (n, x) -> `if`(n = 0, 1, add(E2(n, k)*k!*(2*n - k - 1)!*(-x)^k, k = 0..n)):
seq(print([n], seq(coeff(p(n, x), x, k), k = 0..n)), n = 0..7);
seq(`if`(n = 0, 1, p(n, 1)/(2*(2*n-1)!)), n = 0..14); # check Bernoulli numbers
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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