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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
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

LINKS

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

EXAMPLE

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.

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-Fran├žois Alcover, Aug 04 2014 *)

CROSSREFS

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

Sequence in context: A198062 A226690 A318557 * A273712 A129634 A266825

Adjacent sequences:  A245680 A245681 A245682 * A245684 A245685 A245686

KEYWORD

sign,tabl

AUTHOR

Paul Curtz, Jul 29 2014

STATUS

approved

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Last modified February 18 09:39 EST 2020. Contains 332011 sequences. (Running on oeis4.)