A117440 A cyclically signed version of Pascal's triangle.

1, 1, 1, -1, 2, 1, -1, -3, 3, 1, 1, -4, -6, 4, 1, 1, 5, -10, -10, 5, 1, -1, 6, 15, -20, -15, 6, 1, -1, -7, 21, 35, -35, -21, 7, 1, 1, -8, -28, 56, 70, -56, -28, 8, 1, 1, 9, -36, -84, 126, 126, -84, -36, 9, 1, -1, 10, 45, -120, -210, 252, 210, -120, -45, 10, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

Column k has e.g.f.: (x^k/k!)*(cos(x) + sin(x)).

T(n, k) = binomial(n,k)*(cos(Pi*(n-k)/2) + sin(Pi*(n-k)/2)).

Sum_{k=0..n} T(n, k) = A009545(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = A117441(n) (upward diagonal sums).

G.f.: (1 + x - x*y)/(1 - 2*x*y + x^2*(1 + y^2)). - Stefano Spezia, Mar 10 2024

Example

Triangle begins:
   1;
   1,  1;
  -1,  2,   1;
  -1, -3,   3,   1;
   1, -4,  -6,   4,   1;
   1,  5, -10, -10,   5,   1;
  -1,  6,  15, -20, -15,   6, 1;
  -1, -7,  21,  35, -35, -21, 7, 1;

Mathematica

Table[Binomial[n, k]*(Cos[Pi*(n-k)/2] +Sin[Pi*(n-k)/2]), {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 01 2021 *)

Prog

(Sage) flatten([[binomial(n, k)*( cos(pi*(n-k)/2) + sin(pi*(n-k)/2) ) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2021

Crossrefs

Cf. A007318, A009545 (row sums), A117441 (diagonal sums), A117442 (inverse).

Sequence in context: A094495 A374452 A154926 * A118433 A007318 A108086

Adjacent sequences: A117437 A117438 A117439 * A117441 A117442 A117443

Keyword

easy,sign,tabl

Author

Paul Barry, Mar 16 2006

Status

approved