A051643 Central elements in Parker's partition triangle.

1, 3, 20, 169, 1667, 18084, 208960, 2527074, 31630390, 406680465, 5342750699, 71442850111, 969548468960, 13323571588607, 185072895183632, 2594890728951909, 36681505784903758, 522291180086851188, 7484621370716999785, 107876522368295972285, 1562916545414144667559

Offset: 0

Author

James Sellers

Keywords

• easy
• nonn
• nice

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..90
• R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Crossrefs

Cf. A007042, A047812, A136621.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i
       b(2*n*(n+1), 2*n+1$2):
seq(a(n), n=0..20);  # Alois P. Heinz, May 30 2020

Mathematica

a[n_] := SeriesCoefficient[QBinomial[2(2n+1), 2n+1, q], {q, 0, 2n(n+1)}];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 19 2019 *)

Formula

a(n) = coefficient of q^((m^2-1)/2) = q(2*n*(n+1)) in the q-binomial coefficient [2*m, m] = [2*(2*n+1), 2*n+1], where m = 2*n+1. [Corrected by Petros Hadjicostas, May 30 2020]

a(n) is the number of partitions of 2*n*(n+1) into at most 2*n+1 parts each no bigger than 2*n+1. - Petros Hadjicostas, May 30 2020

Extensions

a(18)-a(20) from Alois P. Heinz, May 30 2020