A294790 Subtract n from partial sums of partial sums of Catalan numbers.

1, 2, 5, 13, 35, 99, 295, 920, 2975, 9892, 33605, 116104, 406615, 1440026, 5147877, 18550573, 67310939, 245716095, 901759951, 3325066997, 12312494463, 45766188949, 170702447075, 638698318851, 2396598337951, 9016444758503, 34003644251207, 128524394659915

Offset: 0

Author

Eric M. Schmidt, Nov 08 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1668
• Christopher Bao, Yunseo Choi, Katelyn Gan, and Owen Zhang, On a Conjecture by Baril, Cerbai, Khalil, and Vajnovszki on Two Restricted Stacks, arXiv:2308.09344 [math.CO], 2023.
• Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
• Giulio Cerbai, Anders Claesson, Luca Ferrari, and Einar Steingrímsson, Sorting with pattern-avoiding stacks: the 132-machine, arXiv:2006.05692 [math.CO], 2020.
• Lapo Cioni and Luca Ferrari, Enumerative Results on the Schröder Pattern Poset, 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.

Crossrefs

Cf. A014140. - Michael De Vlieger, Oct 01 2019

Programs

Maple

a:= proc(n) option remember; `if`(n<3, 1+n^2,
     (6*n*a(n-1)-(9*n-3)*a(n-2)+(4*n-2)*a(n-3))/(n+1))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 01 2019

Mathematica

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 *)

Formula

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