A192654 Alternating partial sums of the Floor-Sqrt transform of Catalan numbers.

1, 0, 1, 1, 2, 4, 7, 13, 24, 45, 84, 158, 298, 563, 1072, 2041, 3905, 7481, 14373, 27665, 53354, 103062, 199398, 386314, 749425, 1455557, 2830158, 5508520, 10731798, 20926411, 40839165, 79761979, 155894688, 304904759, 596729336, 1168571061, 2289723082, 4488979177, 8805149707

Offset

0,5

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

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

a(n) = floor(sqrt(A000108(n))) - a(n-1). - Jon Maiga, Nov 16 2018

Mathematica

Table[Sum[(-1)^(n-k)Floor[Sqrt[Binomial[2k, k]/(k+1)]], {k, 0, n}], {n, 0, 40}]
RecurrenceTable[{a[0] == 1, a[n] == Floor[Sqrt[CatalanNumber[n]]] - a[n - 1]}, a, {n, 30}] (* Jon Maiga, Nov 16 2018 *)

Prog

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

Crossrefs

Cf. A000108 (Catalan numbers).

Sequence in context: A265826 A059633 A088353 * A260668 A371789 A018184

Adjacent sequences: A192651 A192652 A192653 * A192655 A192656 A192657

Keyword

nonn

Author

Emanuele Munarini, Jul 07 2011

Status

approved