A305402 - OEIS

%I #23 Sep 08 2022 08:46:21

%S 1,1,-2,3,-4,2,15,-18,9,-2,105,-120,60,-16,2,945,-1050,525,-150,25,-2,

%T 10395,-11340,5670,-1680,315,-36,2,135135,-145530,72765,-22050,4410,

%U -588,49,-2,2027025,-2162160,1081080,-332640,69300,-10080,1008,-64,2

%N A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)*(1+1/(sqrt(1-u^2)))*exp(p*sqrt(1-u^2)).

%C The function f(u, p) = (1/2)*(1+1/(sqrt(1-u^2))) * exp(p*sqrt(1-u^2)) was found while studying the Fresnel-Kirchhoff and the Rayleigh-Sommerfeld theories of diffraction, see the Meijer link.

%C The Taylor expansion of f(u, p) leads to the number triangle T(n, k), see the example section.

%C Normalization of the triangle terms, dividing the T(n, k) by T(n-k, 0), leads to A084534.

%C The row sums equal A003436, n >= 2, respectively A231622, n >= 1.

%D J. W. Goodman, Introduction to Fourier Optics, 1996.

%D A. Papoulis, Systems and Transforms with Applications in Optics, 1968.

%H Andrew Howroyd, <a href="/A305402/b305402.txt">Rows n=0..50 of triangle, flattened</a>

%H M. J. Bastiaans, <a href="https://doi.org/10.1016/0030-4018(78)90080-9">The Wigner distribution function applied to optical signals and systems</a>, Optics Communications, Vol. 25, nr. 1, pp. 26-30, 1978.

%H H. J. Butterweck, <a href="https://doi.org/10.1364/JOSA.67.000060">General theory of linear, coherent optical data processing systems</a>, Journal of the Optical Society of America, Vol. 67, nr. 1, pp. 60-70, 1977.

%H J. W. Meijer, <a href="/A305402/a305402.pdf">A note on optical diffraction</a>, 1979.

%F T(n, k) = (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!), n > 0 and 0 <= k <= n, T(0, 0) = 1.

%F T(n, k) = (-1)^k*A001147(n-k)*A084534(n, k), n >= 0 and 0 <= k <= n.

%F T(n, k) = 2^(2*(k-n)+1)*A001147(n-k)*A127674(n, n-k), n > 0 and 0 <= k <= n, T(0, 0) = 1.

%F T(n, k) = (-1)^k*(A001497(n, k) + A132062(n, k)), n >= 1, T(0,0) = 1.

%e The first few terms of the Taylor expansion of f(u; p) are:

%e f(u, p) = exp(p) * (1 + (1-2*p) * u^2/4 + (3-4*p+2*p^2) * u^4/16 + (15-18*p+9*p^2-2*p^3) * u^6/96 + (105-120*p+60*p^2-16*p^3+2*p^4) * u^8/768 + ... )

%e The first few rows of the T(n, k) triangle are:

%e n=0:     1

%e n=1:     1,     -2

%e n=2:     3,     -4,    2

%e n=3:    15,    -18,    9,    -2

%e n=4:   105,   -120,   60,   -16,   2

%e n=5:   945,  -1050,  525,  -150,  25,  -2

%e n=6: 10395, -11340, 5670, -1680, 315, -36, 2

%p T := proc(n, k): if n=0 then 1 else (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!) fi: end: seq(seq(T(n, k), k=0..n), n=0..8);

%t Table[If[n==0 && k==0,1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!)], {n, 0, 10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 08 2018 *)

%o (PARI) T(n,k) = {if(n==0, 1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!))}

%o for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 08 2018

%o (Magma) [[n le 0 select 1 else (-1)^k*2^(k-n+1)*Factorial(2*n-k-1)*Binomial(n, k)/Factorial(n-1): k in [0..n]]: n in [1..10]]; // _G. C. Greubel_, Nov 08 2018

%Y Cf. A003436, A231622, A032184, A084534, A127674, A132062, A001497.

%Y Cf. Related to the left hand columns: A001147, A001193, A261065.

%Y Cf. Related to the right hand columns: A280560, A162395, A006011, A040977, A053347, A054334, A266561.

%K sign,easy,tabl

%O 0,3

%A _Johannes W. Meijer_, May 31 2018