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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.
2
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, 17, 17, 13, 5, -5, -13, -17, -17, 0, 17, 17, 47, 13, 47, 17, 17, 0, -31, -31, -107, -73, -13, 13, 73, 107, 31, 31, 0, -31, -31, -355
OFFSET
0,25
COMMENTS
The difference table of the Euler polynomials evaluated at x=1:
1, 1/2, 0, -1/4, 0, 1/2, 0, -17/8, ...
-1/2, -1/2, -1/4, 1/4, 1/2, -1/2, -17/8, 17/8, ...
0, 1/4, 1/2, 1/4; -1, -13/8, 17/4, 107/8, ...
1/4, 1/4, -1/4, -5/4, -5/8, 47/8, 73/8, -355/8, ...
0, -1/2, -1, 5/8 13/2, 13/4, -107/2, -655/8, ...
-1/2, -1/2, 13/8, 47/8, -13/4, -227/4, -227/8, 5687/8, ...
0, 17/8, 17/4, -73/8, -107/2, 227/8, 2957/4, 2957/8, ...
17/8, 17/8, -107/8, -355/8, 655/8, 5687/8, -2957/8, -107125/8, ...
To compute the difference table, take
1, 1/2;
-1/2;
The next term is always half of the sum of the antidiagonals. Hence (-1/2 + 1/2 = 0)
1, 1/2, 0;
-1/2, -1/2;
0;
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.
A198631(n)/A006519(n+1) is an autosequence. See A181722.
Note the main diagonal: 1, -1/2, 1/2, -5/4, 13/2, -227/4, 2957/4, -107125/8, .... (See A212196/A181131.)
This twice the first upper diagonal. The autosequence is of the second kind.
From 0, -1, the algorithm gives A226158(n), full Genocchi numbers, autosequence of the first kind.
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
EXAMPLE
Read by antidiagonals:
1;
-1/2, 1/2;
0, -1/2, 0;
1/4, 1/4, -1/4, -1/4;
0, 1/4, 1/2, 1/4, 0;
-1/2, -1/2, -1/4, 1/4, 1/2, 1/2;
0, -1/2, - 1, -5/4, -1, -1/2, 0;
...
Row sums: 1, 0, -1/2, 0, 1, 0, -17/4, 0, ... = 2*A198631(n+1)/A006519(n+2).
Denominators: 1, 1, 2, 1, 1, 1, 4, 1, ... = A160467(n+2)?
MAPLE
DifferenceTableEulerPolynomials := proc(n) local A, m, k, x;
A := array(0..n, 0..n); x := 1;
for m from 0 to n do for k from 0 to n do A[m, k]:= 0 od od;
for m from 0 to n do A[m, 0] := euler(m, x);
for k from m-1 by -1 to 0 do
A[k, m-k] := A[k+1, m-k-1] - A[k, m-k-1] od od;
LinearAlgebra[Transpose](convert(A, Matrix)) end:
DifferenceTableEulerPolynomials(7); # Peter Luschny, Jul 18 2013
MATHEMATICA
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 *)
PROG
(SageMath)
def DifferenceTableEulerPolynomialsEvaluatedAt1(n) :
@CachedFunction
def ep1(n): # Euler polynomial at x=1
if n < 2: return 1 - n/2
s = add(binomial(n, k)*ep1(k) for k in (0..n-1))
return 1 - s/2
T = matrix(QQ, n)
for m in range(n) : # Compute difference table
T[m, 0] = ep1(m)
for k in range(m-1, -1, -1) :
T[k, m-k] = T[k+1, m-k-1] - T[k, m-k-1]
return T
def A227577_list(m):
D = DifferenceTableEulerPolynomialsEvaluatedAt1(m)
return [D[k, n-k].numerator() for n in range(m) for k in (0..n)]
A227577_list(12) # Peter Luschny, Jul 18 2013
CROSSREFS
KEYWORD
sign
AUTHOR
Paul Curtz, Jul 16 2013
EXTENSIONS
Corrected by Jean-François Alcover, Jul 17 2013
STATUS
approved