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URL: https://oeis.org/A109984

⇱ A109984 - OEIS

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A109984
a(n) = number of steps in all Delannoy paths of length n.
2
0, 5, 44, 321, 2184, 14325, 91860, 580097, 3622928, 22437477, 138049020, 844881345, 5148375192, 31258302933, 189199514532, 1142148091905, 6878977097760, 41347348295877, 248082231062988, 1486116788646977
OFFSET
0,2
COMMENTS
A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).
LINKS
Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
FORMULA
a(n) = sum_{k=0..2n} k*A109983(k).
a(n) = sum_{k=n..2*n} k*binomial(n, 2*n-k)*binomial(k, n).
G.f.: z*(5-z)/(1-6*z+z^2)^(3/2).
Recurrence: (n-1)*(2*n-9)*a(n) = 4*(3*n^2-15*n+7)*a(n-1) - (n-1)*(2*n-7)*a(n-2). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ sqrt(24 + 17*sqrt(2))*(3 + 2*sqrt(2))^n*sqrt(n)/(4*sqrt(Pi)). - Vaclav Kotesovec, Oct 18 2012
EXAMPLE
a(1)=5 because in the 3 (=A001850(1)) Delannoy paths of length 1, namely D, NE and EN, we have altogether five steps.
MAPLE
a:=n->add(k*binomial(n, 2*n-k)*binomial(k, n), k=n..2*n): seq(a(n), n=0..23);
MATHEMATICA
CoefficientList[Series[x*(5-x)/(1-6*x+x^2)^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)
PROG
(Haskell)
a109984 = sum . zipWith (*) [0..] . a109983_row
-- Reinhard Zumkeller, Nov 18 2014
CROSSREFS
Sequence in context: A271118 A268762 A068311 * A247776 A374300 A227640
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jul 07 2005
STATUS
approved