A239223 Number T(n,k) of partitions of n 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, 2, 3, 4, 1, 6, 1, 8, 2, 1, 10, 4, 1, 12, 7, 2, 1, 15, 10, 4, 1, 19, 14, 6, 2, 1, 23, 21, 7, 4, 1, 25, 32, 14, 3, 2, 1, 33, 39, 19, 6, 3, 1, 41, 51, 27, 10, 3, 2, 1, 44, 70, 39, 13, 7, 2, 1, 51, 92, 52, 21, 9, 3, 2, 1, 58, 121, 69, 30, 10, 6, 2, 1, 67, 149

Offset: 1

Author

Alois P. Heinz, Mar 12 2014

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
   1;
   2;
   3;
   4,  1;
   6,  1;
   8,  2,  1;
  10,  4,  1;
  12,  7,  2, 1;
  15, 10,  4, 1;
  19, 14,  6, 2, 1;
  23, 21,  7, 4, 1;
  25, 32, 14, 3, 2, 1;
		

Links

• Alois P. Heinz, Rows n = 1..65, flattened
• Wikipedia, Standard deviation

Crossrefs

Column k=0 gives A238616.

Row sums give A000041.

Maximal index in row n is A140106(n).

Cf. A239228.

Programs

Maple

b:= proc(n, i, m, s, c) `if`(n=0, x^floor(sqrt(s/c-(m/c)^2)),
      `if`(i=1, b(0$2, m+n, s+n, c+n), add(b(n-i*j, i-1,
       m+i*j, s+i^2*j, c+j), j=0..n/i)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0$3)):
seq(T(n), n=1..18);

Mathematica

b[n_, i_, m_, s_, c_] := b[n, i, m, s, c] = If[n==0, x^Floor[Sqrt[s/c - (m/c)^2]], If[i==1, b[0, 0, m+n, s+n, c+n], Sum[b[n-i*j, i-1, m+i*j, s + i^2*j, c+j], {j, 0, n/i}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0, 0, 0]]; Table[T[n], {n, 1, 18}] // Flatten (* Jean-François Alcover, Nov 17 2015, translated from Maple *)