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

Author

Paul Barry, Sep 23 2004

Keywords

• easy
• sign
• tabl

Comments

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

Diagonal sums are A099038.

Examples

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

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.

Crossrefs

Cf. A000984, A099038, A098593,

Programs

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 *)

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

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).