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A051643
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Central elements in Parker's partition triangle.
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5
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1, 3, 20, 169, 1667, 18084, 208960, 2527074, 31630390, 406680465, 5342750699, 71442850111, 969548468960, 13323571588607, 185072895183632, 2594890728951909, 36681505784903758, 522291180086851188, 7484621370716999785, 107876522368295972285, 1562916545414144667559
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OFFSET
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0,2
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LINKS
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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.
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A007042, A047812, A136621.
Sequence in context: A341963 A276315 A145329 * A213377 A357094 A216583
Adjacent sequences: A051640 A051641 A051642 * A051644 A051645 A051646
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KEYWORD
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easy,nonn,nice
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AUTHOR
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James A. Sellers
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EXTENSIONS
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a(18)-a(20) from Alois P. Heinz, May 30 2020
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STATUS
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approved
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