A082143 First subdiagonal of number array A082137.
1, 3, 20, 140, 1008, 7392, 54912, 411840, 3111680, 23648768, 180590592, 1384527872, 10650214400, 82158796800, 635361361920, 4924050554880, 38233804308480, 297374033510400, 2316387208396800, 18067820225495040, 141101072237199360, 1103153837490831360
Offset: 0
Examples
a(0)=(2^(-1)+(0^0)/2)C(1,0)=2*(1/2)=1 (use 0^0=1).
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Programs
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Haskell
a082143 0 = 1 a082143 n = (a000079 $ n - 1) * (a001700 n) -- Reinhard Zumkeller, Jan 15 2015
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Magma
[(2^(n-1) + 0^n/2)*Binomial(2*n+1,n): n in [0..30]]; // G. C. Greubel, Feb 05 2018
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Mathematica
Join[{1}, Table[2^(n-1)* Binomial[2*n+1,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *) -
PARI
for(n=0,30, print1((2^(n-1) + 0^n/2)*Binomial(2*n+1,n), ", ")) \\ G. C. Greubel, Feb 05 2018
Formula
a(n) = (2^(n-1) + 0^n/2)*C(2n+1, n).
Conjecture: (n+1)*a(n) +4*(-2*n-1)*a(n-1)=0. - R. J. Mathar, Oct 19 2014
From Reinhard Zumkeller, Jan 15 2015: (Start)
a(n) = A069720(n+1)/2. (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 64*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)) + 1/7.
Sum_{n>=0} (-1)^n/a(n) = 32*log(2)/27 - 1/9. (End)