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A082145
A subdiagonal of number array A082137.
5
1, 5, 42, 336, 2640, 20592, 160160, 1244672, 9674496, 75246080, 585761792, 4564377600, 35602145280, 277970595840, 2172375244800, 16992801914880, 133035751833600, 1042374243778560, 8173537721057280, 64136851016908800, 503613708419727360, 3956964851869286400
OFFSET
0,2
LINKS
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)
EXAMPLE
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
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
MATHEMATICA
Join[{1}, Table[2^(n-1)* Binomial[2*n+3, n], {n, 1, 30}]] (* G. C. Greubel, Feb 05 2018 *)
PROG
(Magma) [(2^(n-1)+(0^n)/2)*Binomial(2*n+3, n): n in [0..30]]; // Vincenzo Librandi, Oct 30 2014
(PARI) for(n=0, 30, print1((2^(n-1) + 0^n/2)*Binomial(2*n+3, n), ", ")) \\ G. C. Greubel, Feb 05 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Apr 06 2003
STATUS
approved