A245683 - OEIS

%I #17 Mar 23 2017 07:06:18

%S 0,2,1,0,1,1,-6,-3,-1,0,0,-3,-3,-2,-1,50,25,11,4,1,0,0,25,25,18,11,6,

%T 3,-854,-427,-201,-88,-35,-12,-3,0,0,-427,-427,-314,-201,-118,-65,-34,

%U -17,24930,12465,6019,2796,1241,520,201,68,17,0

%N Array T(n,k) read by antidiagonals, where T(0,k) = -A226158(k) and T(n+1,k) = 2*T(n,k+1) - T(n,k).

%C Take T(n,k) = -A226158(k) and its transform via T(n+1,k) = 2*T(n,k+1) - T(n,k):

%C 0,       1,    1,    0,   -1,    0,   3,   0, -17, ...

%C 2,       1,   -1,   -2,    1,    6,  -3, -34, ...     = A230324

%C 0,      -3,   -3,    4,   11,  -12, -65, ...

%C -6,     -3,   11,   18,  -35, -118, ...

%C 0,      25,   25,  -88, -201, ...

%C 50,     25, -201, -314, ...

%C 0,    -427, -427, ...

%C -854, -427, ...

%C 0, ...

%C Every row is alternatively an autosequence of the first kind, see A226158, and of the second kind, see A190339.

%C The second column is twice 1, -3, 25, -427, 12465, ... = (-1)^n*A009843(n) which is in the third column. See A132049(n), numerators of Euler's formula for Pi from the Bernoulli numbers, A243963 and A245244. Hence a link between the Genocchi numbers and Pi.

%C a(n) is the triangle of the increasing antidiagonals.

%e Triangle a(n):

%e    0,

%e    2,  1,

%e    0,  1,  1,

%e   -6, -3, -1,  0,

%e    0, -3, -3, -2, -1,

%e   50, 25, 11,  4,  1,  0,

%e   etc.

%t t[0, 0] = 0; t[0, 1] = 1; t[0, k_] := -k*EulerE[k-1, 0]; t[n_, k_] := t[n, k] = -t[n-1, k] + 2*t[n-1, k+1]; Table[t[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 04 2014 *)

%Y Cf. A226158, A230324, A009843, A132049, A243963, A245244.

%K sign,tabl

%O 0,2

%A _Paul Curtz_, Jul 29 2014