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 A227577 #41 Feb 06 2021 21:49:49
%S A227577 1,-1,1,0,-1,0,1,1,-1,-1,0,1,1,1,0,-1,-1,-1,1,1,1,0,-1,-1,-5,-1,-1,0,
%T A227577 17,17,13,5,-5,-13,-17,-17,0,17,17,47,13,47,17,17,0,-31,-31,-107,-73,
%U A227577 -13,13,73,107,31,31,0,-31,-31,-355
%N A227577 Square array read by antidiagonals, A(n,k) the numerators of the elements of the difference table of the Euler polynomials evaluated at x=1, for n>=0, k>=0.
%C A227577 The difference table of the Euler polynomials evaluated at x=1:
%C A227577 1, 1/2, 0, -1/4, 0, 1/2, 0, -17/8, ...
%C A227577 -1/2, -1/2, -1/4, 1/4, 1/2, -1/2, -17/8, 17/8, ...
%C A227577 0, 1/4, 1/2, 1/4; -1, -13/8, 17/4, 107/8, ...
%C A227577 1/4, 1/4, -1/4, -5/4, -5/8, 47/8, 73/8, -355/8, ...
%C A227577 0, -1/2, -1, 5/8 13/2, 13/4, -107/2, -655/8, ...
%C A227577 -1/2, -1/2, 13/8, 47/8, -13/4, -227/4, -227/8, 5687/8, ...
%C A227577 0, 17/8, 17/4, -73/8, -107/2, 227/8, 2957/4, 2957/8, ...
%C A227577 17/8, 17/8, -107/8, -355/8, 655/8, 5687/8, -2957/8, -107125/8, ...
%C A227577 To compute the difference table, take
%C A227577 1, 1/2;
%C A227577 -1/2;
%C A227577 The next term is always half of the sum of the antidiagonals. Hence (-1/2 + 1/2 = 0)
%C A227577 1, 1/2, 0;
%C A227577 -1/2, -1/2;
%C A227577 0;
%C A227577 The first column (inverse binomial transform) lists the numbers (1, -1/2, 0, 1/4, ..., not in the OEIS; corresponds to A027641/A027642). See A209308 and A060096.
%C A227577 A198631(n)/A006519(n+1) is an autosequence. See A181722.
%C A227577 Note the main diagonal: 1, -1/2, 1/2, -5/4, 13/2, -227/4, 2957/4, -107125/8, .... (See A212196/A181131.)
%C A227577 This twice the first upper diagonal. The autosequence is of the second kind.
%C A227577 From 0, -1, the algorithm gives A226158(n), full Genocchi numbers, autosequence of the first kind.
%C A227577 The difference table of the Bernoulli polynomials evaluated at x=1 is (apart from signs) A085737/A085738 and its analysis by Ludwig Seidel was discussed in the Luschny link. - _Peter Luschny_, Jul 18 2013
%H A227577 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ComputationAndAsymptoticsOfBernoulliNumbers">The computation and asymptotics of the Bernoulli numbers</a>.
%H A227577 OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>
%e A227577 Read by antidiagonals:
%e A227577 1;
%e A227577 -1/2, 1/2;
%e A227577 0, -1/2, 0;
%e A227577 1/4, 1/4, -1/4, -1/4;
%e A227577 0, 1/4, 1/2, 1/4, 0;
%e A227577 -1/2, -1/2, -1/4, 1/4, 1/2, 1/2;
%e A227577 0, -1/2, - 1, -5/4, -1, -1/2, 0;
%e A227577 ...
%e A227577 Row sums: 1, 0, -1/2, 0, 1, 0, -17/4, 0, ... = 2*A198631(n+1)/A006519(n+2).
%e A227577 Denominators: 1, 1, 2, 1, 1, 1, 4, 1, ... = A160467(n+2)?
%p A227577 DifferenceTableEulerPolynomials := proc(n) local A,m,k,x;
%p A227577 A := array(0..n,0..n); x := 1;
%p A227577 for m from 0 to n do for k from 0 to n do A[m,k]:= 0 od od;
%p A227577 for m from 0 to n do A[m,0] := euler(m,x);
%p A227577 for k from m-1 by -1 to 0 do
%p A227577 A[k,m-k] := A[k+1,m-k-1] - A[k,m-k-1] od od;
%p A227577 LinearAlgebra[Transpose](convert(A, Matrix)) end:
%p A227577 DifferenceTableEulerPolynomials(7); # _Peter Luschny_, Jul 18 2013
%t A227577 t[0, 0] = 1; t[0, k_] := EulerE[k, 1]; t[n_, 0] := -t[0, n]; t[n_, k_] := t[n, k] = t[n-1, k+1] - t[n-1, k]; Table[t[n-k, k] // Numerator, {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 18 2013 *)
%o A227577 (Sage)
%o A227577 def DifferenceTableEulerPolynomialsEvaluatedAt1(n) :
%o A227577 @CachedFunction
%o A227577 def ep1(n): # Euler polynomial at x=1
%o A227577 if n < 2: return 1 - n/2
%o A227577 s = add(binomial(n,k)*ep1(k) for k in (0..n-1))
%o A227577 return 1 - s/2
%o A227577 T = matrix(QQ, n)
%o A227577 for m in range(n) : # Compute difference table
%o A227577 T[m,0] = ep1(m)
%o A227577 for k in range(m-1,-1,-1) :
%o A227577 T[k,m-k] = T[k+1,m-k-1] - T[k,m-k-1]
%o A227577 return T
%o A227577 def A227577_list(m):
%o A227577 D = DifferenceTableEulerPolynomialsEvaluatedAt1(m)
%o A227577 return [D[k,n-k].numerator() for n in range(m) for k in (0..n)]
%o A227577 A227577_list(12) # _Peter Luschny_, Jul 18 2013
%Y A227577 Cf. A164555/A027642 in A190339.
%K A227577 sign
%O A227577 0,25
%A A227577 _Paul Curtz_, Jul 16 2013
%E A227577 Corrected by _Jean-François Alcover_, Jul 17 2013