A082143 - OEIS

%I #19 Jan 16 2024 01:39:49

%S 1,3,20,140,1008,7392,54912,411840,3111680,23648768,180590592,

%T 1384527872,10650214400,82158796800,635361361920,4924050554880,

%U 38233804308480,297374033510400,2316387208396800,18067820225495040,141101072237199360,1103153837490831360

%N First subdiagonal of number array A082137.

%H Reinhard Zumkeller, <a href="/A082143/b082143.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (2^(n-1) + 0^n/2)*C(2n+1, n).

%F Conjecture: (n+1)*a(n) +4*(-2*n-1)*a(n-1)=0. - _R. J. Mathar_, Oct 19 2014

%F From _Reinhard Zumkeller_, Jan 15 2015: (Start)

%F a(n) = A000079(n-1) * A001700(n), for n > 0.

%F a(n) = A069720(n+1)/2. (End)

%F From _Amiram Eldar_, Jan 16 2024: (Start)

%F Sum_{n>=0} 1/a(n) = 64*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)) + 1/7.

%F Sum_{n>=0} (-1)^n/a(n) = 32*log(2)/27 - 1/9. (End)

%e a(0)=(2^(-1)+(0^0)/2)C(1,0)=2*(1/2)=1 (use 0^0=1).

%t Join[{1}, Table[2^(n-1)* Binomial[2*n+1,n], {n,1,30}]] (* _G. C. Greubel_, Feb 05 2018 *)

%o (Haskell)

%o a082143 0 = 1

%o a082143 n = (a000079 $ n - 1) * (a001700 n)

%o -- _Reinhard Zumkeller_, Jan 15 2015

%o (PARI) for(n=0,30, print1((2^(n-1) + 0^n/2)*Binomial(2*n+1,n), ", ")) \\ _G. C. Greubel_, Feb 05 2018

%o (Magma) [(2^(n-1) + 0^n/2)*Binomial(2*n+1,n): n in [0..30]]; // _G. C. Greubel_, Feb 05 2018

%Y Cf. A069723, A082144, A082145.

%Y Cf. A000079, A001700, A069720.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Apr 06 2003