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A239005 Signed version of the Seidel triangle for the Euler numbers, read by rows. 4
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; graph; refs; listen; history; text; internal format)
OFFSET

0,9

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..54.

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} C(m,k)*Euler(n-m+k). - Vladimir Kruchinin, Apr 06 2015

EXAMPLE

The triangle begins:

......................1

....................0....1

.................-1..-1....0

...............0...-1,..-2...-2

.............5....5...4....2....0

The difference table 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,...

MATHEMATICA

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 *)

PROG

(Maxima)

T(n, m):=sum(binomial(m, k)*euler(n-m+k), k, 0, m); /* Vladimir Kruchinin, Apr 06 2015 */

CROSSREFS

Unsigned version is A008280.

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

Sequence in context: A257943 A236935 A008280 * A213187 A195710 A200997

Adjacent sequences:  A239002 A239003 A239004 * A239006 A239007 A239008

KEYWORD

sign

AUTHOR

Paul Curtz, Mar 08 2014

STATUS

approved

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Last modified December 7 19:05 EST 2016. Contains 278895 sequences.