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

Offset: 1

Author

Alois P. Heinz, Mar 12 2014

Keywords

• nonn
• tabf

Examples

			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;
		

Links

• Alois P. Heinz, Rows n = 1..100, flattened
• Wikipedia, Standard deviation

Crossrefs

Column k=0 gives A239140.

Row sums give A000009.

Maximal index in row n is A140106(n).

Cf. A239223.

Programs

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 *)