A025272 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5.

1, 1, 1, 2, 6, 17, 50, 150, 458, 1420, 4460, 14165, 45422, 146870, 478366, 1568076, 5169332, 17127466, 57005220, 190503772, 638986740, 2150472248, 7259433880, 24574658829, 83404648534, 283742556262, 967415888822, 3305124188428

Offset

1,4

Links

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Formula

Conjecture: n*a(n) +(n+1)*a(n-1) +10*(-2*n+5)*a(n-2) +2*(2*n-9)*a(n-3) +2*(12*n-67)*a(n-4) +20*(n-7)*a(n-5)=0. - R. J. Mathar, Nov 21 2014

G.f.: 1/2 - sqrt(4*x^4+4*x^3-4*x+1)/2. - Vaclav Kotesovec, Jan 25 2015

Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 2*(2*n-9)*a(n-3) - 4*(n-6)*a(n-4). - Vaclav Kotesovec, Jan 25 2015

Mathematica

nmax = 30; aa = ConstantArray[0, nmax]; aa[[1]] = 1; aa[[2]] = 1; aa[[3]] = 1; aa[[4]] = 2; Do[aa[[n]] = Sum[aa[[k]]*aa[[n-k]], {k, 1, n-1}], {n, 5, nmax}]; aa (* Vaclav Kotesovec, Jan 25 2015 *)
Rest@ CoefficientList[Series[1/2 - Sqrt[4 x^4 + 4 x^3 - 4 x + 1]/2, {x, 0, 28}], x] (* Michael De Vlieger, Jan 02 2020 *)

Prog

(Magma) a:=[1, 1, 1, 2]; for n in [5..30] do Append(~a, &+[a[k]*a[n-k]:k in [1..n-1]] ); end for; a; // Marius A. Burtea, Jan 02 2020

Crossrefs

Sequence in context: A270863 A027914 A098703 * A356783 A148447 A148448

Adjacent sequences: A025269 A025270 A025271 * A025273 A025274 A025275

Keyword

nonn

Author

Clark Kimberling

Status

approved