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

0,2

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.

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

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i
      <n, 0, b(n, i-1, t)+b(n-i, min(i, n-i), t-1)))
    end:
a:= n-> 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 *)

Crossrefs

Cf. A007042, A047812, A136621.

Sequence in context: A341963 A276315 A145329 * A213377 A357094 A216583

Adjacent sequences: A051640 A051641 A051642 * A051644 A051645 A051646

Keyword

easy,nonn,nice

Author

James A. Sellers

Extensions

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

Status

approved