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%I #17 Aug 25 2019 15:31:41
%S 1,7,42,238,1316,7196,39158,212738,1155889,6287015,34249404,186920468,
%T 1022134288,5600420336,30745867316,169116129308,931937277257,
%U 5144687596447,28449040406262,157571572143538,874089046798212
%N Seventh convolution of Schroeder's (second problem) numbers A001003(n), n >= 0.
%H Vincenzo Librandi, <a href="/A111995/b111995.txt">Table of n, a(n) for n = 0..300</a>
%F G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^7.
%F a(n)= (7/n)*Sum_{k=1..n} binomial(n,k)*binomial(n+k+6,k-1).
%F a(n) = 7*hypergeom([1-n, n+8], [2], -1), n >= 1, a(0)=1.
%F a(n) = ((2+sqrt(18))*(4+sqrt(2))^n) + (2-sqrt(18))*(4-sqrt(2))^n)/4 offset 0.
%F a(n) = fourth binomial transform of 1,3,2,6,4,12. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
%F Recurrence: n*(n+7)*a(n) = (7*n^2+37*n+12)*a(n-1) - (7*n^2+19*n-24)*a(n-2) + (n-3)*(n+4)*a(n-3). - _Vaclav Kotesovec_, Oct 18 2012
%F a(n) ~ 7*sqrt(3*sqrt(2)-4)*(99-70*sqrt(2)) * (3+2*sqrt(2))^(n+7)/(32*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 18 2012
%t CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^7, {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))^7) \\ _G. C. Greubel_, Mar 16 2017
%Y Cf. Seventh column of convolution triangle A011117.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 12 2005
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