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 A239005 Signed version of the Seidel triangle for the Euler numbers, read by rows. 5
 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, -32, -16, 0, 0, -61, -122, -178, -224, -256, -272, -272, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0, 0, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS L. Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), pp. 157-187; see Beilage 4 (p. 187). FORMULA a(n) = A057077(n)*A008280(n) by rows. a(n) is the increasing antidiagonals of the difference table of A155585(n). Central column of triangle: A099023(n). Right main diagonal of triangle: A155585(n) (see A009006(n)). Left main diagonal of triangle: A122045(n). 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).] 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 EXAMPLE The triangle T(n,k) begins:                       1                     0   1                  -1  -1   0                 0  -1  -2  -2               5   5   4   2   0              ... The array read as a table, A(n,k) = T(n+k, k), starts:      1,    1,    0,   -2,    0,   16,    0, -272,    0, ...      0,   -1,   -2,    2,   16,  -16, -272,  272, ...     -1,   -1,    4,   14,  -32, -256,  544, ...      0,    5,   10,  -46, -224,  800, ...      5,    5,  -56, -178, 1024, ...      0,  -61, -122, 1202, ...    -61,  -61, 1324, ...      0, 1385, ...   1385, ...   ... 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 MATHEMATICA t[0, 0] = 1; t[n_, m_] /; n

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Last modified June 14 07:10 EDT 2021. Contains 345018 sequences. (Running on oeis4.)