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A241066 Array t(n,k) = k^(2n)*(k^(2n)-1)*BernoulliB(2n)/(2n), n>=1, k>=2, absolute values read by ascending antidiagonals.

%I A241066 #9 Feb 16 2025 08:33:21
%S A241066 1,2,6,16,54,20,272,2106,544,50,7936,179334,66560,3250,105,353792,
%T A241066 26414586,17895424,968750,13986,196,22368256,5957217414,8329625600,
%U A241066 635781250,8637840,48020,336,1903757312,1906398972666,5937093935104,722480468750,11754617616,54925276,139776,540
%N A241066 Array t(n,k) = k^(2n)*(k^(2n)-1)*BernoulliB(2n)/(2n), n>=1, k>=2, absolute values read by ascending antidiagonals.
%C A241066 For any integers n and k, the ratio k^(2n)*(k^(2n)-1)*B(2n)/(2n) is always an integer.
%C A241066 Row 1 is A002415 = 4-D pyramidal numbers,
%C A241066 Row 2 and following rows are not in the OEIS,
%C A241066 Column 1 is A000182 = Tangent numbers,
%C A241066 Column 2 is A047681,
%C A241066 Column 3 is A047682,
%C A241066 Column 4 is A047683,
%C A241066 Column 5 and following columns are not in the OEIS.
%H A241066 MathWorld, <a href="https://mathworld.wolfram.com/BernoulliNumber.html">Bernoulli Number</a>
%H A241066 Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>
%e A241066 Array begins:
%e A241066    1,        6,         20,           50,            105, ...
%e A241066    2,       54,        544,         3250,          13986, ...
%e A241066   16,     2106,      66560,       968750,        8637840, ...
%e A241066 272,   179334,   17895424,    635781250,    11754617616, ...
%e A241066 7936, 26414586, 8329625600, 722480468750, 27698169542400, ...
%e A241066 etc.
%t A241066 nmax = 8; t[n_, k_] := k^(2*n)*(k^(2*n)-1)*BernoulliB[2*n]/(2*n); Table[t[n-k+2, k] // Abs, {n, 1, nmax}, {k, 2, n+1}] // Flatten
%Y A241066 Cf. A000182, A002415, A047681, A047682, A047683.
%K A241066 nonn,tabl
%O A241066 1,2
%A A241066 _Jean-François Alcover_, Apr 16 2014