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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

%I

%S 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 17,17,13,5,-5,-13,-17,-17,0,17,17,47,13,47,17,17,0,-31,-31,-107,-73,

%U -13,13,73,107,31,31,0,-31,-31,-355

%N 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 The difference table of the Euler polynomials evaluated at x=1:

%C 1, 1/2, 0, -1/4, 0, 1/2, 0, -17/8,...

%C -1/2, -1/2, -1/4, 1/4, 1/2, -1/2, -17/8, 17/8,...

%C 0, 1/4, 1/2, 1/4; -1, -13/8, 17/4, 107/8,...

%C 1/4, 1/4, -1/4, -5/4, -5/8, 47/8, 73/8, -355/8,...

%C 0, -1/2, -1, 5/8 13/2, 13/4, -107/2, -655/8,...

%C -1/2, -1/2, 13/8, 47/8, -13/4, -227/4, -227/8, 5687/8,...

%C 0, 17/8, 17/4, -73/8, -107/2, 227/8, 2957/4, 2957/8,...

%C 17/8, 17/8, -107/8, -355/8, 655/8, 5687/8, -2957/8, -107125/8.

%C To compute the difference table take

%C 1, 1/2,

%C -1/2,

%C The next term is ever the halfsum of the antidiagonals. Hence (-1/2+1/2=0)

%C 1, 1/2, 0,

%C -1/2, -1/2,

%C 0.

%C 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.

%C A198631(n)/A006519(n+1) is an autosequence. See A181722.

%C Note the main diagonal: 1, -1/2, 1/2, -5/4, 13/2, -227/4, 2957/4, -107125/8,... . (See A212196/A181131)

%C This the double of the first upper diagonal. The autosequence is of the second kind.

%C From 0, -1 ,the algorithm gives A226158(n), full Genocchi numbers, autosequence of the first kind.

%C 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 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ComputationAndAsymptoticsOfBernoulliNumbers">The computation and asymptotics of the Bernoulli numbers</a>.

%e Read by antidiagonals:

%e 1

%e -1/2, 1/2,

%e 0, -1/2, 0,

%e 1/4, 1/4, -1/4, -1/4,

%e 0, 1/4, 1/2, 1/4, 0,

%e -1/2, -1/2, -1/4, 1/4, 1/2, 1/2,

%e 0, -1/2, -1, -5/4, -1, -1/2, 0.

%e Row sum: 1, 0, -1/2, 0, 1, 0, -17/4, 0, = 2*A198631(n+1)/A006519(n+2).

%e Denominators: 1, 1, 2, 1, 1, 1, 4, 1,... = A160467(n+2) ?.

%p DifferenceTableEulerPolynomials := proc(n) local A,m,k,x;

%p A := array(0..n,0..n); x := 1;

%p for m from 0 to n do for k from 0 to n do A[m,k]:= 0 od od;

%p for m from 0 to n do A[m,0] := euler(m,x);

%p for k from m-1 by -1 to 0 do

%p A[k,m-k] := A[k+1,m-k-1] - A[k,m-k-1] od od;

%p LinearAlgebra[Transpose](convert(A, Matrix)) end:

%p DifferenceTableEulerPolynomials(7); # _Peter Luschny_, Jul 18 2013

%t 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-Fran├žois Alcover_, Jul 18 2013 *)

%o (Sage)

%o def DifferenceTableEulerPolynomialsEvaluatedAt1(n) :

%o @CachedFunction

%o def ep1(n): # Euler polynomial at x=1

%o if n < 2: return 1 - n/2

%o s = add(binomial(n,k)*ep1(k) for k in (0..n-1))

%o return 1 - s/2

%o T = matrix(QQ, n)

%o for m in range(n) : # Compute difference table

%o T[m,0] = ep1(m)

%o for k in range(m-1,-1,-1) :

%o T[k,m-k] = T[k+1,m-k-1] - T[k,m-k-1]

%o return T

%o def A227577_list(m):

%o D = DifferenceTableEulerPolynomialsEvaluatedAt1(m)

%o return [D[k,n-k].numerator() for n in range(m) for k in (0..n)]

%o A227577_list(12) # _Peter Luschny_, Jul 18 2013

%Y Cf. A164555/A027642 in A190339.

%K sign

%O 0,25

%A _Paul Curtz_, Jul 16 2013

%E Corrected by Jean-Francois Alcover.

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Last modified August 24 03:20 EDT 2019. Contains 326260 sequences. (Running on oeis4.)