A294790 - OEIS

%I #17 Aug 23 2023 10:18:56

%S 1,2,5,13,35,99,295,920,2975,9892,33605,116104,406615,1440026,5147877,

%T 18550573,67310939,245716095,901759951,3325066997,12312494463,

%U 45766188949,170702447075,638698318851,2396598337951,9016444758503,34003644251207,128524394659915

%N Subtract n from partial sums of partial sums of Catalan numbers.

%H Alois P. Heinz, <a href="/A294790/b294790.txt">Table of n, a(n) for n = 0..1668</a>

%H Christopher Bao, Yunseo Choi, Katelyn Gan, and Owen Zhang, <a href="https://arxiv.org/abs/2308.09344">On a Conjecture by Baril, Cerbai, Khalil, and Vajnovszki on Two Restricted Stacks</a>, arXiv:2308.09344 [math.CO], 2023.

%H Giulio Cerbai, Anders Claesson, and Luca Ferrari, <a href="https://arxiv.org/abs/1907.08142">Stack sorting with restricted stacks</a>, arXiv:1907.08142 [cs.DS], 2019.

%H Giulio Cerbai, Anders Claesson, Luca Ferrari, and Einar Steingrímsson, <a href="https://arxiv.org/abs/2006.05692">Sorting with pattern-avoiding stacks: the 132-machine</a>, arXiv:2006.05692 [math.CO], 2020.

%H Lapo Cioni and Luca Ferrari, <a href="https://dx.doi.org/10.1007/978-3-319-58631-1_5">Enumerative Results on the Schröder Pattern Poset</a>, In: Dennunzio A., Formenti E., Manzoni L., Porreca A. (eds) Cellular Automata and Discrete Complex Systems, AUTOMATA 2017, Lecture Notes in Computer Science, vol 10248. See p. 65.

%F a(n) = A014140(n) - n.

%p a:= proc(n) option remember; `if`(n<3, 1+n^2,

%p      (6*n*a(n-1)-(9*n-3)*a(n-2)+(4*n-2)*a(n-3))/(n+1))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 01 2019

%t MapIndexed[#1 - First[#2] + 1 &, CoefficientList[Series[1/(1 - x)^2*(1 - Sqrt[1 - 4 x])/(2 x), {x, 0, 27}], x]] (* _Michael De Vlieger_, Oct 01 2019 *)

%Y Cf. A014140. - _Michael De Vlieger_, Oct 01 2019

%K nonn

%O 0,2

%A _Eric M. Schmidt_, Nov 08 2017