This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A285864 #25 Jun 26 2025 16:12:22
%S A285864 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,
%T A285864 0,32,0,-56,0,14,-7,1,-128,0,128,0,-112,0,56,-8,1,0,-384,0,128,0,-336,
%U A285864 0,24,-9,1,2560,0,-384,0,320,0,-112,0,30,-10,1
%N 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).
%C A285864 The denominator triangle b(n,m) is given in A285865.
%C A285864 a(n,m)/b(n,m) = B(2;n,m) is the d = 2 instance of the fractional d-family of triangles B(d;n,m) = binomial(n,m)*d^(n-m)*B(n-m), for d >= 1. They are the coefficient triangles of generalized Bernoulli polynomials PB(d;n,x) = Sum_{m=0..n} B(d;n,m)*x^m for n >= 0.
%C A285864 {PB(d;n,x)}_{n>=0} has e.g.f. EB(d;x,z) := Sum_{n>=0} PB(d;n,x)*z^n = d*z*exp(x*z)/(exp(d*z)-1). B(d;n,m) is a Sheffer triangle of the Appell type for each d, denoted by (d*z/(exp(d*z - 1)), z).
%C A285864 PB(d;n,x) gives a (trivial) generalization of the Bernoulli polynomials with coefficients given in A196838/A196839 (rising powers of x), and this is PB(1;n,x).
%C A285864 The polynomials PB(d;n,x) appear in the generalized Faulhaber formula for sums of powers of arithmetic progressions SP(n,m) := Sum_{j=0..m} (a + d*j)^n, n >= 0, m >= 0, d >= 1, a = 0 for d = 1 and a from the smallest positive restricted residue system modulo d >= 2. For this Faulhaber formula see a comment in A285863, where they are named B(d;n,x).
%C A285864 The row sums of the rational triangle B(2;n,m) give A157779(n)/A141459(n). The alternating row sums are given in A285866/A141459(n).
%F A285864 a(n,m) = numerator(binomial(n, m)*2^(n-m)*B(n-m)), with the Bernoulli numbers B(k) = A027641(k)/A027642(k).
%F A285864 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.
%e A285864 The triangle a(n,m) begins:
%e A285864 n\m 0 1 2 3 4 5 6 7 8 9 10 ...
%e A285864 0: 1
%e A285864 1: -1 1
%e A285864 2: 2 -2 1
%e A285864 3: 0 2 -3 1
%e A285864 4: -8 0 4 -4 1
%e A285864 5: 0 -8 0 20 -5 1
%e A285864 6: 32 0 -8 0 10 -6 1
%e A285864 7: 0 32 0 -56 0 14 -7 1
%e A285864 8: -128 0 128 0 -112 0 56 -8 1
%e A285864 9: 0 -384 0 128 0 -336 0 24 -9 1
%e A285864 10: 2560 0 -384 0 320 0 -112 0 30 -10 1
%e A285864 ...
%e A285864 The rational triangle B(2;n,m) = a(n,m)/A285865(n,m) begins:
%e A285864 n\m 0 1 2 3 4 5 6 7 8 9 10 ...
%e A285864 0: 1
%e A285864 1: -1 1
%e A285864 2: 2/3 -2 1
%e A285864 3: 0 2 -3 1
%e A285864 4: -8/15 0 4 -4 1
%e A285864 5: 0 -8/3 0 20/3 -5 1
%e A285864 6: 32/21 0 -8 0 10 -6 1
%e A285864 7: 0 32/3 0 -56/3 0 14 -7 1
%e A285864 8: -128/15 0 128/3 0 -112/3 0 56/3 -8 1
%e A285864 9: 0 -384/5 0 128 0 -336/5 0 24 -9 1
%e A285864 10: 2560/33 0 -384 0 320 0 -112 0 30 -10 1
%e A285864 ...
%p A285864 T := d -> (n,m) -> numer(binomial(n, m)*d^(n-m)*bernoulli(n-m)):
%p A285864 for n from 0 to 10 do seq(T(2)(n,k),k=0..n) od; # _Peter Luschny_, May 04 2017
%t A285864 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 *)
%o A285864 (PARI) T(n, m) = numerator(binomial(n, m)*2^(n - m)*bernfrac(n - m));
%o A285864 for(n=0, 20, for(m=0, n, print1(T(n, m),", ");); print();) \\ _Indranil Ghosh_, May 06 2017
%o A285864 (Python)
%o A285864 from sympy import binomial, bernoulli
%o A285864 def T(n, m): return (binomial(n, m) * (-2)**(n - m) * bernoulli(n - m)).numerator
%o A285864 for n in range(21): print([T(n, m) for m in range(n + 1)]) # _Indranil Ghosh_, May 06 2017
%Y A285864 Cf. A027641/A027642, A157779/A141459, A196838/A196839, A285863, A285865.
%K A285864 sign,easy,tabl,frac
%O A285864 0,4
%A A285864 _Wolfdieter Lang_, May 03 2017