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A051643 Central elements in Parker's partition triangle.

%I A051643 #23 Jul 02 2025 16:01:58
%S A051643 1,3,20,169,1667,18084,208960,2527074,31630390,406680465,5342750699,
%T A051643 71442850111,969548468960,13323571588607,185072895183632,
%U A051643 2594890728951909,36681505784903758,522291180086851188,7484621370716999785,107876522368295972285,1562916545414144667559
%N A051643 Central elements in Parker's partition triangle.
%H A051643 Alois P. Heinz, <a href="/A051643/b051643.txt">Table of n, a(n) for n = 0..90</a>
%H A051643 R. K. Guy, <a href="http://www.jstor.org/stable/2324467">Parker's permutation problem involves the Catalan numbers</a>, Amer. Math. Monthly 100 (1993), 287-289.
%F A051643 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]
%F A051643 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
%p A051643 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i
%p A051643       <n, 0, b(n, i-1, t)+b(n-i, min(i, n-i), t-1)))
%p A051643     end:
%p A051643 a:= n-> b(2*n*(n+1), 2*n+1$2):
%p A051643 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 30 2020
%t A051643 a[n_] := SeriesCoefficient[QBinomial[2(2n+1), 2n+1, q], {q, 0, 2n(n+1)}];
%t A051643 Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Aug 19 2019 *)
%Y A051643 Cf. A007042, A047812, A136621.
%K A051643 easy,nonn,nice
%O A051643 0,2
%A A051643 _James Sellers_
%E A051643 a(18)-a(20) from _Alois P. Heinz_, May 30 2020