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A335323
First lower diagonal of Parker's triangle A047812.
1
0, 1, 3, 7, 11, 18, 26, 38, 52, 73, 97, 131, 172, 227, 293, 381, 486, 623, 788, 998, 1251, 1571, 1954, 2432, 3006, 3714, 4561, 5600, 6838, 8345, 10139, 12306, 14879, 17973, 21633, 26011, 31181, 37334, 44579, 53170, 63257, 75171, 89130, 105554, 124750, 147269
OFFSET
1,3
COMMENTS
Apparently, this sequence was originally intended to be A7043 (now A007043), but for some reason it was crossed out on p. 4 of the annotated copy of Guy's 1992 preprint.
a(n) is the number of partitions of (n-2)*(n+1) into at most n parts each no bigger than n. Thus, a(n) is the coefficient of q^((n-2)*(n+1)) in the q-binomial coefficient [2*n, n].
LINKS
R. K. Guy, Parker's permutation problem involves the Catalan numbers, preprint, 1992. (Annotated scanned copy)
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.
EXAMPLE
a(1) = 0 because it does not make sense to talk about the partitions of (1-2)*(1+1) = -2.
a(2) = 1 because we have only the empty partition for (2-2)*(2+1) = 0.
a(3) = 3 because we have the following partitions of (3-2)*(3+1) = 4 into no more than 3 parts each no bigger than 3: 1+3 = 1+1+2 = 2+2.
a(4) = 7 because we have the following partitions of (4-2)*(4+1) = 10 into no more than 4 parts each no bigger than 4: 2+4+4 = 3+3+4 = 1+1+4+4 = 1+2+3+4 = 1+3+3+3 = 2+2+2+4 = 2+2+3+3.
The PARI function partitions((n-2)*(n+1), n, n) can generate these partitions.
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<0
or t*i<n, 0, b(n, i-1, t)+b(n-i, min(i, n-i), t-1)))
end:
a:= n-> b((n-2)*(n+1), n$2):
seq(a(n), n=1..50); # Alois P. Heinz, May 31 2020
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[n < 0 || t i < n, 0, b[n, i - 1, t] + b[n - i, Min[i, n - i], t - 1]]];
a[n_] := b[(n-2)(n+1), n, n];
Array[a, 50] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)
PROG
(PARI) T(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
for(n=1, 43, print1(T(n, n-2), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Petros Hadjicostas, May 31 2020
STATUS
approved