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A239228
Number T(n,k) of partitions of n into distinct parts with standard deviation σ in the half-open interval [k,k+1); triangle T(n,k), n>=1, 0<=k<=max(0,floor(n/2)-1), read by rows.
3
1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 2, 2, 1, 1, 4, 3, 1, 1, 2, 4, 3, 2, 1, 2, 4, 5, 2, 1, 1, 2, 5, 6, 2, 2, 1, 1, 5, 8, 4, 2, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 7, 9, 7, 4, 2, 1, 1, 2, 6, 12, 9, 4, 3, 1, 1, 2, 5, 15, 11, 6, 3, 2, 1, 1, 2, 6, 16
OFFSET
1,3
LINKS
EXAMPLE
Triangle T(n,k) begins:
1;
1;
2;
1, 1;
2, 1;
2, 1, 1;
2, 2, 1;
1, 3, 1, 1;
3, 2, 2, 1;
1, 4, 3, 1, 1;
2, 4, 3, 2, 1;
2, 4, 5, 2, 1, 1;
MAPLE
b:= proc(n, i, m, s, c) `if`(n>i*(i+1)/2, 0, `if`(n=0,
x^floor(sqrt(s/c-(m/c)^2)), b(n, i-1, m, s, c)+
`if`(i>n, 0, b(n-i, i-1, m+i, s+i^2, c+1))))
end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0$3)):
seq(T(n), n=1..20);
MATHEMATICA
b[n_, i_, m_, s_, c_] := If[n > i*(i + 1)/2, 0, If[n == 0, x^Floor[Sqrt[ s/c - (m/c)^2]], b[n, i - 1, m, s, c] + If[i > n, 0, b[n - i, i - 1, m + i, s + i^2, c + 1]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, Exponent[#, x]}]&[b[n, n, 0, 0, 0]];
Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, May 22 2018, translated from Maple *)
CROSSREFS
Column k=0 gives A239140.
Row sums give A000009.
Maximal index in row n is A140106(n).
Cf. A239223.
Sequence in context: A268318 A309414 A007421 * A346080 A103921 A115623
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Mar 12 2014
STATUS
approved