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Row sums of triangle A155856.
4

%I #26 Dec 23 2024 11:33:41

%S 1,2,6,23,107,590,3786,27821,230869,2137978,21873854,245151555,

%T 2987967551,39358156310,557259550034,8440866957273,136211005966889,

%U 2333068710452146,42276699542130166,808068680469402095,16248405328930779027,342877404288485770718,7576652528705018522906

%N Row sums of triangle A155856.

%C For positive n, a(n) equals the permanent of the n X n matrix with 2's along the main diagonal and the upper diagonal, and 1's everywhere else. - _John M. Campbell_, Jul 09 2011

%H Vincenzo Librandi, <a href="/A155857/b155857.txt">Table of n, a(n) for n = 0..200</a>

%H Veronica Bitonti, Bishal Deb, and Alan D. Sokal, <a href="https://arxiv.org/abs/2412.10214">Thron-type continued fractions (T-fractions) for some classes of increasing trees</a>, arXiv:2412.10214 [math.CO], 2024. See p. 58.

%F G.f.: 1/(1 -x -x/(1 -x -x/(1 -x -2*x/(1 -x -2*x/(1 -x -3*x/(1 -x -3*x/(1 - ... (continued fraction);

%F a(n) = Sum_{k=0..n} binomial(2*n-k, k)*(n-k)!.

%F a(n) = Sum_{k=0..n} binomial(n+k, 2*k)*k!. - _Paul Barry_, May 28 2009

%F a(n) = (n+1)*a(n-1) -(n-3)*a(n-2) -a(n-3). - _R. J. Mathar_, Nov 15 2012

%F a(n) ~ exp(2) * n!. - _Vaclav Kotesovec_, Feb 08 2014

%t Table[Sum[Binomial[2*n-k,k]*(n-k)!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Feb 08 2014 *)

%o (Sage) [sum(binomial(2*n-k, k)*factorial(n-k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Jun 05 2021

%Y Cf. A155856.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Jan 29 2009