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

1,2

Links

Alois P. Heinz, Rows n = 1..65, flattened

Wikipedia, Standard deviation

Example

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;

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 *)

Crossrefs

Column k=0 gives A238616.

Row sums give A000041.

Maximal index in row n is A140106(n).

Cf. A239228.

Sequence in context: A124406 A225650 A340087 * A143771 A364255 A366283

Adjacent sequences: A239220 A239221 A239222 * A239224 A239225 A239226

Keyword

nonn,tabf

Author

Alois P. Heinz, Mar 12 2014

Status

approved