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

1,5

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.

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

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 *)

Crossrefs

Sequence in context: A166292 A010374 A307788 * A217333 A089372 A036671

Adjacent sequences: A025270 A025271 A025272 * A025274 A025275 A025276

Keyword

nonn

Author

Clark Kimberling

Status

approved