A305402 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)).

1, 1, -2, 3, -4, 2, 15, -18, 9, -2, 105, -120, 60, -16, 2, 945, -1050, 525, -150, 25, -2, 10395, -11340, 5670, -1680, 315, -36, 2, 135135, -145530, 72765, -22050, 4410, -588, 49, -2, 2027025, -2162160, 1081080, -332640, 69300, -10080, 1008, -64, 2

Offset

0,3

Comments

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.

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

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

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

References

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

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

Links

Andrew Howroyd, Rows n=0..50 of triangle, flattened

M. J. Bastiaans, The Wigner distribution function applied to optical signals and systems, Optics Communications, Vol. 25, nr. 1, pp. 26-30, 1978.

H. J. Butterweck, General theory of linear, coherent optical data processing systems, Journal of the Optical Society of America, Vol. 67, nr. 1, pp. 60-70, 1977.

J. W. Meijer, A note on optical diffraction, 1979.

Formula

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.

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

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.

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

Example

The first few terms of the Taylor expansion of f(u; p) are:
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 + ... )
The first few rows of the T(n, k) triangle are:
n=0:     1
n=1:     1,     -2
n=2:     3,     -4,    2
n=3:    15,    -18,    9,    -2
n=4:   105,   -120,   60,   -16,   2
n=5:   945,  -1050,  525,  -150,  25,  -2
n=6: 10395, -11340, 5670, -1680, 315, -36, 2

Maple

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);

Mathematica

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 *)

Prog

(PARI) T(n, k) = {if(n==0, 1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 08 2018
(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

Crossrefs

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

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

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

Sequence in context: A117009 A204842 A103300 * A213394 A363774 A358606

Adjacent sequences: A305399 A305400 A305401 * A305403 A305404 A305405

Keyword

sign,easy,tabl

Author

Johannes W. Meijer, May 31 2018

Status

approved