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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.
3
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
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
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Mar 12 2014
STATUS
approved