A239228 - OEIS

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.

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

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OFFSET

1,3

LINKS

Alois P. Heinz, Rows n = 1..100, flattened

Wikipedia, Standard deviation

EXAMPLE

Triangle T(n,k) begins:

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

Adjacent sequences: A239225 A239226 A239227 * A239229 A239230 A239231

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Mar 12 2014

STATUS

approved