A104858 Partial sums of the little Schroeder numbers (A001003).

1, 2, 5, 16, 61, 258, 1161, 5440, 26233, 129282, 648141, 3294864, 16943733, 87983106, 460676625, 2429478144, 12893056497, 68802069506, 368961496469, 1987323655056, 10746633315501, 58321460916482, 317537398625945

Offset: 0

Author

Emeric Deutsch, Apr 24 2005

Keywords

• nonn

Comments

The subsequence of primes begins: 2, 5, 61, no more through a(30). [Jonathan Vos Post, Feb 12 2010]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Crossrefs

Cf. A001003, A005408, A059993.

Programs

Maple

G:=(1+z-sqrt(1-6*z+z^2))/4/z/(1-z): Gser:=series(G,z=0,29): 1,seq(coeff(Gser,z^n),n=1..27);

Mathematica

CoefficientList[Series[(1+x-Sqrt[1-6*x+x^2])/4/x/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

Formula

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

Recurrence: (n+1)*a(n) = (7*n-2)*a(n-1) - (7*n-5)*a(n-2) + (n-2)*a(n-3). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ sqrt(24 + 17*sqrt(2))*(3 + 2*sqrt(2))^n/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

Define a triangle T(n,1) = T(n,n) = 1 for n >= 1 and all other elements by T(r,c) = T(r,c-1) + T(r-1,c-1) + T(r-1,c). Its second column is A005408, its third column is A059993, and the sum of all terms in its row n is a(n-1). - J. M. Bergot, Dec 01 2012