OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
FORMULA
Binomial transform of A100071 starting [1, 2, 6, 12, 30, ...].
Inverse binomial transform of A037965 starting [1, 4, 18, 80, 350, ...].
a(n) = (n-1)! * [x^(n-1)] exp(x)*((1 + 2*x)*BesselI(0, 2*x) + 2*x*BesselI(1, 2*x)) for n>0, a(0) = 0. - Peter Luschny, Aug 26 2012
D-finite with recurrence (n-1)*a(n) = 3*(n-1)*a(n-1) +(n+1)*a(n-2) -3*(n-3)*a(n-3). - R. J. Mathar, Nov 09 2021
G.f.: x*(1-x)/((1-3*x)*sqrt((1+x)*(1-3*x))). - G. C. Greubel, May 28 2024
EXAMPLE
a(3) = 11 = (1, 2, 1) dot (1, 2, 6) = (1 + 4 + 6), where A100071 = (1, 2, 6, 12, 30, ...).
a(3) = 11 = (1, -2, 1) dot (1, 4, 18) = (1 - 8 + 18). - Gary W. Adamson, Nov 10 2007
MATHEMATICA
a[n_]:= a[n]= Sum[(-1)^(n-k-1)*Binomial[n-1, k]*(k+1)*Binomial[2*k, k], {k, 0, n-1}];
Table[a[n], {n, 40}] (* G. C. Greubel, May 28 2024 *)
PROG
(Magma)
A134757:= func< n | (&+[(-1)^(n-k-1)*(k+1)^2*Binomial(n-1, k)*Catalan(k) : k in [0..n-1]]) >;
[A134757(n): n in [1..40]]; // G. C. Greubel, May 28 2024
(SageMath)
def A134757(n): return sum((-1)^(n-k-1)*(k+1)*binomial(n-1, k)*binomial( 2*k, k) for k in range(n))
[A134757(n) for n in range(1, 41)] # G. C. Greubel, May 28 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Nov 08 2007
STATUS
approved