cp's OEIS Frontend

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

%I A239005 #62 Feb 22 2021 04:04:12
%S A239005 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 A239005 -32,-16,0,0,-61,-122,-178,-224,-256,-272,-272,1385,1385,1324,1202,
%U A239005 1024,800,544,272,0,0,1385,2770,4094,5296,6320,7120,7664,7936,7936
%N A239005 Signed version of the Seidel triangle for the Euler numbers, read by rows.
%H A239005 L. Seidel, <a href="http://publikationen.badw.de/de/003384831/pdf/CC%20BY">Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), pp. 157-187; see Beilage 4 (p. 187).
%F A239005 a(n) = A057077(n)*A008280(n) by rows.
%F A239005 a(n) is the increasing antidiagonals of the difference table of A155585(n).
%F A239005 Central column of triangle: A099023(n).
%F A239005 Right main diagonal of triangle: A155585(n) (see A009006(n)).
%F A239005 Left main diagonal of triangle: A122045(n).
%F A239005 T(n,m) = Sum_{k=0..n} binomial(m,k)*Euler(n-m+k) for 0 <= m <= n. - _Vladimir Kruchinin_, Apr 06 2015 [The summation only needs to go from k=0 to k=m because of binomial(m,k).]
%F A239005 T(n,k) = (-1)^n*A236935(n-k,k) for 0 <= k <= n, where the latter is read as a square array. - _Petros Hadjicostas_, Feb 21 2021
%e A239005 The triangle T(n,k) begins:
%e A239005                       1
%e A239005                     0   1
%e A239005                  -1  -1   0
%e A239005                 0  -1  -2  -2
%e A239005               5   5   4   2   0
%e A239005              ...
%e A239005 The array read as a table, A(n,k) = T(n+k, k), starts:
%e A239005      1,    1,    0,   -2,    0,   16,    0, -272,    0, ...
%e A239005      0,   -1,   -2,    2,   16,  -16, -272,  272, ...
%e A239005     -1,   -1,    4,   14,  -32, -256,  544, ...
%e A239005      0,    5,   10,  -46, -224,  800, ...
%e A239005      5,    5,  -56, -178, 1024, ...
%e A239005      0,  -61, -122, 1202, ...
%e A239005    -61,  -61, 1324, ...
%e A239005      0, 1385, ...
%e A239005   1385, ...
%e A239005   ...
%e A239005 For the above table, we have A(n,k) = (-1)^(n+k)*A236935(n,k) for n, k >= 0. It has joint e.g.f. 2*exp(-x)/(1 + exp(-2*(x+y))). - _Petros Hadjicostas_, Feb 21 2021
%t A239005 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 A239005 (Maxima)
%o A239005 T(n,m):=sum(binomial(m,k)*euler(n-m+k),k,0,m); /* _Vladimir Kruchinin_, Apr 06 2015 */
%o A239005 (PARI) a(n) = 2^n*2^(n+1)*(subst(bernpol(n+1, x), x, 3/4) - subst(bernpol(n+1, x), x, 1/4))/(n+1) /* A122045 */
%o A239005 T(n, k) = (-1)^n*sum(i=0, k, (-1)^i*binomial(k, i)*a(n-i)) /* _Petros Hadjicostas_, Feb 21 2021 */
%o A239005 /* Second PARI program (same a(n) for A122045 as above) */
%o A239005 T(n, k) = sum(i=0, k, binomial(k, i)*a(n-k+i)) /* _Petros Hadjicostas_, Feb 21 2021 */
%Y A239005 Unsigned version is A008280.
%Y A239005 Cf. A008281, A099023, A108040, A122045, A155585, A236935.
%K A239005 sign,tabl
%O A239005 0,9
%A A239005 _Paul Curtz_, Mar 08 2014