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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 (list; graph; refs; listen; history; text; internal format)
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 ever the halfsum of the antidiagonals. Hence (-1/2+1/2=0)

1,     1/2, 0,

-1/2, -1/2,

0.

The first column (inverse binomial transform) is 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 the double of 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

LINKS

Table of n, a(n) for n=0..58.

Peter Luschny, The computation and asymptotics of the Bernoulli numbers.

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 sum: 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

(Sage)

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

Cf. A164555/A027642 in A190339.

Sequence in context: A220962 A201292 A086464 * A196840 A162298 A196755

Adjacent sequences:  A227574 A227575 A227576 * A227578 A227579 A227580

KEYWORD

sign

AUTHOR

Paul Curtz, Jul 16 2013

EXTENSIONS

Corrected by Jean-Francois Alcover.

STATUS

approved

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Last modified October 21 14:16 EDT 2014. Contains 248377 sequences.