A117435 Triangle related to exp(x)*cos(2*x).

1, 0, 1, -4, 0, 1, 0, -12, 0, 1, 16, 0, -24, 0, 1, 0, 80, 0, -40, 0, 1, -64, 0, 240, 0, -60, 0, 1, 0, -448, 0, 560, 0, -84, 0, 1, 256, 0, -1792, 0, 1120, 0, -112, 0, 1, 0, 2304, 0, -5376, 0, 2016, 0, -144, 0, 1, -1024, 0, 11520, 0, -13440, 0, 3360, 0, -180, 0, 1

Offset

0,4

Comments

Diagonals correspond to rows of A117438.

Links

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

Formula

Number triangle whose k-th column has e.g.f. (x^k/k!)*cos(2x);

T(n, k) = binomial(n,k) * (-4)^((n-k)/2) * (1+(-1)^(n-k))/2.

Sum_{k=0..n} T(n, k) = A006495(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = i^n * ((1+(-1)^n)/2) * (2*floor(n/2) + 1). - G. C. Greubel, Jun 01 2021

Example

Triangle begins:
    1;
    0,   1;
   -4,   0,   1;
    0, -12,   0,   1;
   16,   0, -24,   0,   1;
    0,  80,   0, -40,   0, 1;
  -64,   0, 240,   0, -60, 0, 1;

Mathematica

T[n_, k_]:= Binomial[n, k]*(2*I)^(n-k)*(1+(-1)^(n+k))/2;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 01 2021 *)

Prog

(Sage) flatten([[binomial(n, k)*(2*i)^(n-k)*(1+(-1)^(n+k))/2 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2021

Crossrefs

Cf. A006495 (row sums), A117411, A117436 (inverse), A117438.

Sequence in context: A244530 A372722 A271424 * A282252 A268367 A117436

Adjacent sequences: A117432 A117433 A117434 * A117436 A117437 A117438

Keyword

easy,sign,tabl

Author

Paul Barry, Mar 16 2006

Status

approved