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A165433
A transform of the double factorial numbers A001147.
1
1, 1, 2, 3, 7, 14, 39, 97, 308, 897, 3139, 10304, 38997, 140893, 570002, 2230599, 9567979, 40091222, 181203603, 805962157, 3819522284, 17912075229, 88646095447, 435959031488, 2245454002137, 11530035000169, 61627679281154
OFFSET
0,3
COMMENTS
Hankel transform is A000178.
LINKS
FORMULA
G.f.: 1/(1-x-x^2-2x^4/(1-x-5x^2-12x^4/(1-x-9x^2-30x^4/(1-x-13x^2-56x^4/(1-.... (continued fraction);
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*(2k)!/(k!*2^k).
Conjecture: 2*a(n) -3*a(n-1) +(3-2*n)*a(n-2) +(2*n-3)*a(n-3)=0. - R. J. Mathar, Nov 14 2011
G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 29 2013
a(n) ~ 2^(-1/2) * exp(sqrt(n)/2 - n/2 + 1/16) * n^(n/2) * (1 + 121/(192*sqrt(n))). - Vaclav Kotesovec, Apr 18 2024
MAPLE
a:=proc(n) add(binomial(n-k, k)*factorial(2*k)/(factorial(k)*2^k), k=0..floor(n/2)) end proc: seq(a(n), n=0..30); # Muniru A Asiru, Oct 20 2018
MATHEMATICA
Table[Sum[Binomial[n-k, k]*(2*k)!/(k!*2^k), {k, 0, Floor[n/2]}], {n, 0, 30}] (* G. C. Greubel, Oct 20 2018 *)
PROG
(PARI) vector(30, n, n--; sum(k=0, floor(n/2), binomial(n-k, k)*(2*k)!/(k!*2^k))) \\ G. C. Greubel, Oct 20 2018
(Magma) [(&+[Binomial(n-k, k)*Factorial(2*k)/(Factorial(k)*2^k): k in [0.. Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Oct 20 2018
CROSSREFS
Sequence in context: A006785 A274538 A113182 * A238728 A271718 A222626
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 18 2009
STATUS
approved