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A239005 Signed version of the Seidel triangle for the Euler numbers, read by rows. 4

%I

%S 1,0,1,-1,-1,0,0,-1,-2,-2,5,5,4,2,0,0,5,10,14,16,16,-61,-61,-56,-46,

%T -32,-16,0,0,-61,-122,-178,-224,-256,-272,-272,1385,1385,1324,1202,

%U 1024,800,544,272,0,0,1385,2770,4094,5296,6320,7120,7664,7936,7936

%N Signed version of the Seidel triangle for the Euler numbers, read by rows.

%D L. Seidel, Ueber eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

%F a(n) = A057077(n)*A008280(n) by rows.

%F a(n) is the increasing antidiagonals of the difference table of A155585(n).

%F Central column of triangle: A099023(n).

%F Right main diagonal of triangle: A155585(n) (see A009006(n)).

%F Left main diagonal of triangle: A122045(n).

%F T(n,m) = Sum_{k=0..n} C(m,k)*Euler(n-m+k). - _Vladimir Kruchinin_, Apr 06 2015

%e The triangle begins:

%e ......................1

%e ....................0....1

%e .................-1..-1....0

%e ...............0...-1,..-2...-2

%e .............5....5...4....2....0

%e The difference table starts:

%e 1, 1, 0, -2, 0, 16, 0, -272, 0,...

%e 0, -1, -2, 2, 16, -16, -272, 272,...

%e -1, -1, 4, 14, -32, -256, 544,...

%e 0, 5, 10, -46, -224, 800,...

%e 5, 5, -56, -178, 1024,...

%e 0, -61, -122, 1202,...

%e -61, -61, 1324,...

%e 0, 1385,...

%e 1385,...

%t t[0, 0] = 1; t[n_, m_] /; n<m || m<0 = 0; t[n_, m_] := t[n, m] = Sum[t[n-1, n-k], {k, m}]; Table[r = (-1)^Floor[n/2]*Table[t[n, m], {m, 0, n}]; If[EvenQ[n], Reverse[r], r], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Dec 30 2014 *)

%o (Maxima)

%o T(n,m):=sum(binomial(m,k)*euler(n-m+k),k,0,m); /* _Vladimir Kruchinin_, Apr 06 2015 */

%Y Unsigned version is A008280.

%Y Cf. A108040, A008281, A155585, A099023, A122045.

%K sign

%O 0,9

%A _Paul Curtz_, Mar 08 2014

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Last modified July 21 19:25 EDT 2019. Contains 325199 sequences. (Running on oeis4.)