|
|
A057552
|
|
a(n) = Sum_{k=0..n} C(2k+2,k).
|
|
28
|
|
|
1, 5, 20, 76, 286, 1078, 4081, 15521, 59279, 227239, 873885, 3370029, 13027729, 50469889, 195892564, 761615284, 2965576714, 11563073314, 45141073924, 176423482324, 690215089744, 2702831489824, 10593202603774, 41550902139550, 163099562175850, 640650742051802
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
LINKS
|
|
|
FORMULA
|
G.f.: 1/2*(2*x+(1-4*x)^(1/2)-1)/(1-4*x)^(1/2)/x^2/(-1+x). - Vladeta Jovovic, Sep 10 2003
D-finite with recurrence: n*(n+2)*a(n) = (5*n^2+8*n+2)*a(n-1) - 2*(n+1)*(2*n+1)*a(n-2). - Vaclav Kotesovec, Oct 11 2012
a(n) = Sum_{k=1..n+1} k*A000108(k) = Sum_{k=1..n+1} A001791(k) = (A000108(n+1) * (4*n + 6 - (n+2)*hypergeom([1,-n-1], [-n-1/2], 1/4]) - 1)/2.
|
|
MAPLE
|
a:= n->add(binomial(2*j+2, j), j=0..n): seq(a(n), n=0..24); # Zerinvary Lajos, Oct 25 2006
|
|
MATHEMATICA
|
Table[Sum[Binomial[2k+2, k], {k, 0, n}], {n, 0, 20}]
(* or *)
Table[SeriesCoefficient[1/2*(2*x+(1-4*x)^(1/2)-1)/(1-4*x)^(1/2)/x^2/(-1+x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 11 2012 *)
Table[(CatalanNumber[n + 1] (4 n + 6 - (n + 2) Hypergeometric2F1[1, -n-1, -n-1/2, 1/4]) - 1)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)
|
|
PROG
|
(PARI) a(n) = sum(k=0, n, binomial(2*k+2, k)); \\ Michel Marcus, Oct 04 2016
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|