A155857 Row sums of triangle A155856.

1, 2, 6, 23, 107, 590, 3786, 27821, 230869, 2137978, 21873854, 245151555, 2987967551, 39358156310, 557259550034, 8440866957273, 136211005966889, 2333068710452146, 42276699542130166, 808068680469402095, 16248405328930779027, 342877404288485770718, 7576652528705018522906

Offset: 0

Author

Paul Barry, Jan 29 2009

Keywords

• nonn
• easy

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.

Crossrefs

Cf. A155856.

Programs

Mathematica

Table[Sum[Binomial[2*n-k,k]*(n-k)!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Feb 08 2014 *)

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

Formula

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);

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

a(n) = Sum_{k=0..n} binomial(n+k, 2*k)*k!. - Paul Barry, May 28 2009

a(n) = (n+1)*a(n-1) -(n-3)*a(n-2) -a(n-3). - R. J. Mathar, Nov 15 2012

a(n) ~ exp(2) * n!. - Vaclav Kotesovec, Feb 08 2014