OFFSET
0,4
COMMENTS
Conjecture: Starting with n=1, a(n) is the main diagonal of the array defined as T(1,1) = 1, T(i,j) = 0 if i<1 or j<1, T(n,k) = T(n,k-2) + T(n,k-1) -2*T(n-1,k-1) + T(n-1,k) + T(n-2,k). - Gerald McGarvey, Oct 07 2008
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = Sum_{j=0..n} (-1)^(j-1)*C(2*n-j-1, n-j)*C(2*(j-1), j-1). - corrected by Peter Bala, Aug 19 2014
a(n) = n*A055392(n), n>1.
a(n) = binomial(2*n-2, n-1)*Hypergeometric([-(n-1), 1/2], [2-2*n], -4) with a(0) = 0, a(1) = 1. - G. C. Greubel, Jan 11 2022
MATHEMATICA
a[n_]:= Sum[(-1)^j*Binomial[2*j, j]*Binomial[2*n-j-2, n-j-1], {j, 0, n-1}];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jan 11 2022 *)
PROG
(PARI) a(n) = sum(j=0, n, (-1)^(j-1)*binomial(2*n-j-1, n-j)*binomial(2*(j-1), j-1)); \\ Michel Marcus, Aug 19 2014
(Sage)
def a(n): return n if (n<2) else binomial(2*n-2, n-1)*simplify( hypergeometric([-(n-1), 1/2], [2-2*n], -4) )
[a(n) for n in (0..30)] # G. C. Greubel, Jan 11 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 09 2005
STATUS
approved