A111994 - OEIS

%I #13 Mar 17 2017 00:47:17

%S 1,6,33,176,930,4908,25954,137712,733539,3922834,21060099,113481504,

%T 613619332,3328768344,18112655748,98833261600,540705999621,

%U 2965360687518,16299708148901,89784615643728,495545294427558

%N Sixth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.

%H Vincenzo Librandi, <a href="/A111994/b111994.txt">Table of n, a(n) for n = 0..200</a>

%F G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^6.

%F a(n)= (6/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+5,k-1).

%F a(n) = 6*hypergeom([1-n, n+7], [2], -1), n>=1, a(0)=1.

%F Recurrence: n*(n+6)*a(n) = (7*n^2+30*n+5)*a(n-1) - (7*n^2+12*n-22)*a(n-2) + (n-3)*(n+3)*a(n-3). - _Vaclav Kotesovec_, Oct 18 2012

%F a(n) ~ 3*sqrt(3*sqrt(2)-4)*(58-41*sqrt(2)) * (3+2*sqrt(2))^(n+6)/(16*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 18 2012

%t CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^6, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 18 2012 *)

%o (PARI) x='x+O('x^50); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^6) \\ _G. C. Greubel_, Mar 16 2017

%Y Cf. Sixth column of convolution triangle A011117.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Sep 12 2005