A086872
Triangle T(n, k) read by rows; given by [1, 2, 3, 4, 5, 6, ..] DELTA [1, 4, 9, 16, 25, 36, ...] where DELTA is the operator defined in A084938.
Original entry on oeis.org
1, 1, 1, 3, 8, 5, 15, 75, 121, 61, 105, 840, 2478, 3128, 1385, 945, 11025, 51030, 115350, 124921, 50521, 10395, 166320, 1105335, 3859680, 7365633, 7158128, 2702765
Offset: 0
Triangle begins:
1;
1, 1;
3, 8, 5;
15, 75, 121, 61;
105, 840, 2478, 3128, 1385;
945, 11025, 51030, 115350, 124921, 50521;
10395, 166320, 1105335, 3859680, 7365633, 7158128, 2702765 ; ...
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)).
Original entry on oeis.org
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
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
- J. W. Goodman, Introduction to Fourier Optics, 1996.
- A. Papoulis, Systems and Transforms with Applications in Optics, 1968.
- 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.
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[[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
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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);
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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 *)
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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
Showing 1-2 of 2 results.
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