A225749 - OEIS

%I #14 Feb 16 2025 08:33:19

%S 1,0,1,-1,0,1,0,-1,0,1,1,0,-1,0,5,0,1,0,-1,0,691,-1,0,1,0,-691,0,7,0,

%T -1,0,691,0,-1,0,3617,1,0,-691,0,1,0,-3617,0,43867,0,691,0,-1,0,3617,

%U 0,-43867,0,174611,-691,0,1,0,-3617,0,43867,0,-174611,0,854513,0,-1,0,3617,0,-43867,0,174611,0,-77683,0,236364091

%N Triangle T(n,k) giving numerator of integral_{x=0..1} B(n,x)*B(k,x) dx, B = Bernoulli polynomial, n >= 1, 1 <= k <= n.

%H Vincenzo Librandi, <a href="/A225749/b225749.txt">Rows n = 0..100, flattened</a>

%H NIST Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/24.13#i">Bernoulli Polynomials</a>

%H Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BernoulliPolynomial.html">Bernoulli Polynomial</a>

%e Triangle begins:

%e 1;

%e 0, 1;

%e -1, 0, 1;

%e 0, -1, 0, 1;

%e 1, 0, -1, 0, 5;

%e 0, 1, 0, -1, 0, 691;

%e etc.

%t t[n_, k_] := (-1)^(n - 1)*k!*n!/(k + n)!*BernoulliB[k + n]; Table[t[n, k] // Numerator, {n, 1, 12}, {k, 1, n}] // Flatten

%Y Cf. A225750 (denominators)

%K sign,frac,tabl

%O 0,15

%A _Jean-François Alcover_, May 14 2013