A285864 Triangle read by rows: a(n,m) = numerator(binomial(n,m)*2^(n-m)*B(n-m)) with B(k) the Bernoulli numbers A027641(k)/A027642(k).
1, -1, 1, 2, -2, 1, 0, 2, -3, 1, -8, 0, 4, -4, 1, 0, -8, 0, 20, -5, 1, 32, 0, -8, 0, 10, -6, 1, 0, 32, 0, -56, 0, 14, -7, 1, -128, 0, 128, 0, -112, 0, 56, -8, 1, 0, -384, 0, 128, 0, -336, 0, 24, -9, 1, 2560, 0, -384, 0, 320, 0, -112, 0, 30, -10, 1
Offset: 0
Examples
The triangle a(n,m) begins: n\m 0 1 2 3 4 5 6 7 8 9 10 ... 0: 1 1: -1 1 2: 2 -2 1 3: 0 2 -3 1 4: -8 0 4 -4 1 5: 0 -8 0 20 -5 1 6: 32 0 -8 0 10 -6 1 7: 0 32 0 -56 0 14 -7 1 8: -128 0 128 0 -112 0 56 -8 1 9: 0 -384 0 128 0 -336 0 24 -9 1 10: 2560 0 -384 0 320 0 -112 0 30 -10 1 ... The rational triangle B(2;n,m) = a(n,m)/A285865(n,m) begins: n\m 0 1 2 3 4 5 6 7 8 9 10 ... 0: 1 1: -1 1 2: 2/3 -2 1 3: 0 2 -3 1 4: -8/15 0 4 -4 1 5: 0 -8/3 0 20/3 -5 1 6: 32/21 0 -8 0 10 -6 1 7: 0 32/3 0 -56/3 0 14 -7 1 8: -128/15 0 128/3 0 -112/3 0 56/3 -8 1 9: 0 -384/5 0 128 0 -336/5 0 24 -9 1 10: 2560/33 0 -384 0 320 0 -112 0 30 -10 1 ...
Programs
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Maple
T := d -> (n,m) -> numer(binomial(n, m)*d^(n-m)*bernoulli(n-m)): for n from 0 to 10 do seq(T(2)(n,k),k=0..n) od; # Peter Luschny, May 04 2017
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Mathematica
T[n_, m_]:=Numerator[Binomial[n, m]*2^(n - m)*BernoulliB[n - m]]; Table[T[n, m], {n, 0, 20}, {m, 0, n}] // Flatten (* Indranil Ghosh, May 06 2017 *)
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PARI
T(n, m) = numerator(binomial(n, m)*2^(n - m)*bernfrac(n - m)); for(n=0, 20, for(m=0, n, print1(T(n, m),", ");); print();) \\ Indranil Ghosh, May 06 2017
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Python
from sympy import binomial, bernoulli def T(n, m): return (binomial(n, m) * (-2)**(n - m) * bernoulli(n - m)).numerator for n in range(21): print([T(n, m) for m in range(n + 1)]) # Indranil Ghosh, May 06 2017
Formula
a(n,m) = numerator(binomial(n, m)*2^(n-m)*B(n-m)), with the Bernoulli numbers B(k) = A027641(k)/A027642(k).
E.g.f.s of the rational column sequences {B(2;n, m)}_{n>=0} are Ecol(m, x) = (2*x/(exp(2*x) - 1))*x^m/m! (Sheffer property). Here the numerators of column m are numerator([x^m/m!] Ecol(m, x)), m >= 0.
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