A192654 - OEIS

%I #13 Sep 08 2022 08:45:57

%S 1,0,1,1,2,4,7,13,24,45,84,158,298,563,1072,2041,3905,7481,14373,

%T 27665,53354,103062,199398,386314,749425,1455557,2830158,5508520,

%U 10731798,20926411,40839165,79761979,155894688,304904759,596729336,1168571061,2289723082,4488979177,8805149707

%N Alternating partial sums of the Floor-Sqrt transform of Catalan numbers.

%H G. C. Greubel, <a href="/A192654/b192654.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*floor(sqrt(binomial(2*k,k)/(k+1))).

%F a(n) = floor(sqrt(A000108(n))) - a(n-1). - _Jon Maiga_, Nov 16 2018

%t Table[Sum[(-1)^(n-k)Floor[Sqrt[Binomial[2k,k]/(k+1)]],{k,0,n}],{n,0,40}]

%t RecurrenceTable[{a[0] == 1, a[n] == Floor[Sqrt[CatalanNumber[n]]] - a[n - 1]}, a, {n, 30}] (* _Jon Maiga_, Nov 16 2018 *)

%o (Maxima) makelist(sum((-1)^(n-k)*floor(sqrt(binomial(2*k,k)/(k+1))), k,0,n), n,0,24);

%o (PARI) vector(40, n, n--; sum(k=0,n, (-1)^(n-k)*floor( sqrt(binomial(2*k, k)/(k+1))))) \\ _G. C. Greubel_, Nov 16 2018

%o (Magma) [(&+[(-1)^(n-k)*Floor(Sqrt(Binomial(2*k,k)/(k+1))): k in [0..n]]) : n in [0..40]]; // _G. C. Greubel_, Nov 16 2018

%o (Sage) [sum((-1)^(n-k+1)*floor(sqrt(binomial(2*k,k)/(k+1))) for k in range(n)) for n in (1..40)] # _G. C. Greubel_, Nov 16 2018

%Y Cf. A000108 (Catalan numbers).

%K nonn

%O 0,5

%A _Emanuele Munarini_, Jul 07 2011