A245683 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).

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, 3, -854, -427, -201, -88, -35, -12, -3, 0, 0, -427, -427, -314, -201, -118, -65, -34, -17, 24930, 12465, 6019, 2796, 1241, 520, 201, 68, 17, 0

Offset: 0

Author

Paul Curtz, Jul 29 2014

Keywords

• sign
• tabl

Comments

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

0, 1, 1, 0, -1, 0, 3, 0, -17, ...

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

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

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

0, 25, 25, -88, -201, ...

50, 25, -201, -314, ...

0, -427, -427, ...

-854, -427, ...

0, ...

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

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.

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

Examples

			Triangle a(n):
   0,
   2,  1,
   0,  1,  1,
  -6, -3, -1,  0,
   0, -3, -3, -2, -1,
  50, 25, 11,  4,  1,  0,
  etc.
		

Crossrefs

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

Programs

Mathematica

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 *)