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A335323
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First lower diagonal of Parker's triangle A047812.
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1
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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
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OFFSET
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1,3
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COMMENTS
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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].
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LINKS
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EXAMPLE
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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.
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MAPLE
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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):
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MATHEMATICA
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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];
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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