A112521 Sequence related to NOR bracketings.

0, 1, 0, 6, 4, 60, 84, 700, 1440, 8910, 23100, 120120, 360360, 1684956, 5552064, 24302520, 85101456, 357502860, 1302562404, 5333981796, 19947127200, 80408748420, 305922388200, 1221485157360, 4701015343440, 18664243014300

Offset: 0

Author

Paul Barry, Sep 09 2005

Keywords

• easy
• nonn

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

Crossrefs

Cf. A055392.

Programs

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 *)

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

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