A025273 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5, starting 1,0,1,1.

1, 0, 1, 1, 2, 5, 12, 29, 72, 182, 466, 1207, 3158, 8334, 22158, 59299, 159614, 431838, 1173710, 3203244, 8774780, 24118522, 66497316, 183858411, 509670494, 1416231616, 3944027402, 11006186760, 30772507128, 86191006746, 241815195292, 679488418879

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

The binomial transform of A025250(n+1) is A025273(n+2). - Paul Barry, May 11 2005

Links

• Robert Israel, Table of n, a(n) for n = 1..2140
• Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.

Programs

Maple

f:= gfun:-rectoproc({(n+1)*a(n) +2*(-2*n+1)*a(n-1) +4*(n-2)*a(n-2) +2*(-2*n+7)*a(n-3) +4*(n-5)*a(n-4)=0, a(0)=1,a(1)=0,a(2)=1,a(3)=1},a(n),remember):
map(f, [$0..50]); # Robert Israel, Nov 02 2016
# alternative
A025273 := proc(n)
    option remember ;
    if n < 5 then
        op(n,[1,0,1,1]) ;
    else
        add( procname(i)*procname(n-i),i=1..n-1) ;
    end if;
end proc:
seq(A025273(n),n=1..20) ; # R. J. Mathar, Jan 13 2025

Mathematica

nmax = 30; aa = ConstantArray[0,nmax]; aa[[1]] = 1; aa[[2]] = 0; aa[[3]] = 1; aa[[4]] = 1; Do[aa[[n]] = Sum[aa[[k]]*aa[[n-k]],{k,1,n-1}],{n,5,nmax}]; aa (* Vaclav Kotesovec, Jan 25 2015 *)
CoefficientList[Series[(1-Sqrt[1-4*x+4*x^2-4*x^3+4*x^4])/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2015 *)

Formula

G.f. (with offset 0 instead of 1): (1-sqrt(1-4*x+4*x^2-4*x^3+4*x^4))/(2*x). - Paul Barry, May 11 2005

Conjecture: (with offset 0 instead of 1) (n+1)*a(n) +2*(-2*n+1)*a(n-1) +4*(n-2)*a(n-2) +2*(-2*n+7)*a(n-3) +4*(n-5)*a(n-4)=0. - R. J. Mathar, Nov 24 2012

Conjecture follows from the differential equation 4*x^3-3*x^2+2*x-1+(-4*x^4+2*x^3-2*x+1)*g(x)+(4*x^5-4*x^4+4*x^3-4*x^2+x)*g'(x)=0 satisfied by the g.f. - Robert Israel, Nov 02 2016