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

0,2

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.

Formula

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

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 *)

Crossrefs

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

Sequence in context: A308273 A126221 A107086 * A234643 A089846 A258450

Adjacent sequences: A294787 A294788 A294789 * A294791 A294792 A294793

Keyword

nonn

Author

Eric M. Schmidt, Nov 08 2017

Status

approved