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Seventh convolution of Schroeder's (second problem) numbers A001003(n), n >= 0.
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%I #25 Sep 07 2024 03:22:13

%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) = 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) my(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

%E Incorrect formula removed by _Jason Yuen_, Sep 07 2024