A099037 Triangle of diagonals of symmetric Krawtchouk matrices.

1, 1, -1, 1, 0, 1, 1, 3, -3, -1, 1, 8, -12, 8, 1, 1, 15, -20, 20, -15, -1, 1, 24, -15, 0, -15, 24, 1, 1, 35, 21, -105, 105, -21, -35, -1, 1, 48, 112, -336, 420, -336, 112, 48, 1, 1, 63, 288, -672, 756, -756, 672, -288, -63, -1, 1, 80, 585, -960, 420, 0, 420, -960, 585, 80, 1, 1, 99, 1045, -825, -1980, 4620, -4620, 1980, 825, -1045, -99, -1

Offset

0,8

Comments

Row sums have e.g.f. BesselI(0,2*x) (A000984 with interpolated zeros).

Diagonal sums are A099038.

References

P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum walks, Contemporary Mathematics, 287 2001, pp. 83-96.

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

P. Feinsilver and R. Fitzgerald, The Spectrum of Symmetric Krawtchouk Matrices, Linear Algebra and Its Applications, Vol. 235 (1996), pp. 121-139.

Formula

Triangle T(n, k) = if(k<=n, C(n, k)*Sum_{i=0..n} (-1)^i*C(k, i)C(n-k, k-i), 0).

Triangle T(n, k) = Sum_{j=0..n} (-1)^(n-j)*C(n,j)*C(j,k)*C(k,j-k) = C(n,k)*A098593(n,k).

Example

Triangle begins as:
1.
1, -1.
1,  0,  1.
1,  3, -3,  1.
1,  8, -12, 8, 1. ...

Mathematica

T[n_, k_]:= If[k <= n, Binomial[n, k]*Sum[(-1)^j*Binomial[k, j]*Binomial[n - k, k - j], {j, 0, n}], 0]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 31 2017 *)

Prog

(PARI) {T(n, k) = binomial(n, k)*sum(j=0, n, (-1)^j*binomial(k, j)*binomial(n-k, k-j))};
for(n=0, 20, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 31 2017

Crossrefs

Cf. A000984, A099038, A098593,

Sequence in context: A287290 A287981 A213660 * A340934 A271706 A172108

Adjacent sequences: A099034 A099035 A099036 * A099038 A099039 A099040

Keyword

easy,sign,tabl

Author

Paul Barry, Sep 23 2004

Status

approved