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%I #16 Jul 15 2023 05:41:35
%S 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,
%T 24,1,1,35,21,-105,105,-21,-35,-1,1,48,112,-336,420,-336,112,48,1,1,
%U 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
%N Triangle of diagonals of symmetric Krawtchouk matrices.
%C Row sums have e.g.f. BesselI(0,2*x) (A000984 with interpolated zeros).
%C Diagonal sums are A099038.
%D P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum walks, Contemporary Mathematics, 287 2001, pp. 83-96.
%H G. C. Greubel, <a href="/A099037/b099037.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H P. Feinsilver and R. Fitzgerald, <a href="https://doi.org/10.1016/0024-3795(94)00123-5">The Spectrum of Symmetric Krawtchouk Matrices</a>, Linear Algebra and Its Applications, Vol. 235 (1996), pp. 121-139.
%F 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).
%F 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).
%e Triangle begins as:
%e 1.
%e 1, -1.
%e 1, 0, 1.
%e 1, 3, -3, 1.
%e 1, 8, -12, 8, 1. ...
%t 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 *)
%o (PARI) {T(n, k) = binomial(n, k)*sum(j=0,n, (-1)^j*binomial(k, j)*binomial(n-k, k-j))};
%o for(n=0,20, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Dec 31 2017
%Y Cf. A000984, A099038, A098593,
%K easy,sign,tabl
%O 0,8
%A _Paul Barry_, Sep 23 2004
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