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) is the increasing antidiagonals of the difference table of A155585(n).
Central column of triangle: A099023(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<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 */
(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 */
T(n, k) = (-1)^n*sum(i=0, k, (-1)^i*binomial(k, i)*a(n-i)) /* Petros Hadjicostas, Feb 21 2021 */
/* Second PARI program (same a(n) for A122045 as above) */
T(n, k) = sum(i=0, k, binomial(k, i)*a(n-k+i)) /* Petros Hadjicostas, Feb 21 2021 */
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul Curtz, Mar 08 2014
STATUS
approved