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

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

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