VOOZH about

URL: https://oeis.org/A192654

⇱ A192654 - OEIS

login
A192654
Alternating partial sums of the Floor-Sqrt transform of Catalan numbers.
1
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
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) a(n) = sum(k=0, n, (-1)^(n-k)*sqrtint(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
(SageMath) [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
KEYWORD
nonn
AUTHOR
Emanuele Munarini, Jul 07 2011
STATUS
approved