A236407 a(n) = 2*Sum_{k=0..n-1} C(n-1,k)*C(n+k,k) + n.

0, 3, 10, 41, 196, 1007, 5342, 28821, 157192, 864155, 4780018, 26572097, 148321356, 830764807, 4666890950, 26283115053, 148348809232, 838944980531, 4752575891162, 26964373486425, 153196621856212, 871460014012703, 4962895187697070, 28292329581548741

Offset: 0

Author

N. J. A. Sloane, Jan 31 2014

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• J. Brzozowski, M. Szykula, Large Aperiodic Semigroups, arXiv:1401.0157 [cs.FL], 2013 (Tables 1, 2).

Crossrefs

Cf. A002003.

Programs

Mathematica

Table[2*Sum[Binomial[n-1,k]*Binomial[n+k,k],{k,0,n-1}]+n,{n,0,20}] (* Vaclav Kotesovec, Feb 14 2014 *)
Flatten[{0,Table[n+2*Hypergeometric2F1[1-n,1+n,1,-1],{n,1,20}]}] (* Vaclav Kotesovec, Feb 14 2014 *)

PARI

for(n=0,25, print1(n + 2*sum(k=0,n-1, binomial(n-1,k) * binomial(n+k,k)), ", ")) \\ G. C. Greubel, Jun 01 2017

Formula

a(n) = A002003(n) + n.

Conjecture: n*(n-3)*a(n) -4*(2*n-1)*(n-3)*a(n-1) +2*(7*n^2-28*n+20)*a(n-2) -4*(n-1)*(2*n-7)*a(n-3) +(n-1)*(n-4)*a(n-4)=0. - R. J. Mathar, Feb 01 2014

Recurrence: (n-2)*n*(2*n^2 - 8*n + 7)*a(n) = (14*n^4 - 88*n^3 + 189*n^2 - 158*n + 39)*a(n-1) - (14*n^4 - 80*n^3 + 153*n^2 - 112*n + 24)*a(n-2) + (n-3)*(n-1)*(2*n^2 - 4*n + 1)*a(n-3). - Vaclav Kotesovec, Feb 14 2014

a(n) ~ 2^(-3/4) * (3+2*sqrt(2))^n / sqrt(Pi*n). - Vaclav Kotesovec, Feb 14 2014