A082145 A subdiagonal of number array A082137.
1, 5, 42, 336, 2640, 20592, 160160, 1244672, 9674496, 75246080, 585761792, 4564377600, 35602145280, 277970595840, 2172375244800, 16992801914880, 133035751833600, 1042374243778560, 8173537721057280, 64136851016908800, 503613708419727360, 3956964851869286400
Offset: 0
Examples
a(0) = ( 2^(-1)+(0^0)/2 )*C(3,0) = ( 1/2+1/2 )*1 = 1 (use 0^0 = 1). - clarified by _Jon Perry_, Oct 29 2014
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Magma
[(2^(n-1)+(0^n)/2)*Binomial(2*n+3, n): n in [0..30]]; // Vincenzo Librandi, Oct 30 2014
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Maple
Z:=(1-3*z-sqrt(1-4*z))/sqrt(1-4*z)/64: Zser:=series(Z, z=0, 32): seq(coeff(Zser*2^(n+1), z, n), n=4..23); # Zerinvary Lajos, Jan 01 2007
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Mathematica
Join[{1}, Table[2^(n-1)* Binomial[2*n+3,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+3,n), ", ")) \\ G. C. Greubel, Feb 05 2018
Formula
a(n) = ( 2^(n-1) + (0^n)/2 )*binomial(2*n+3, n).
(n+3)*a(n) +2*(-7*n-13)*a(n-1) +24*(2*n+1)*a(n-2)=0. - R. J. Mathar, Oct 29 2014
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 37/7 - 208*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = 296*log(2)/27 - 61/9. (End)