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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.
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%I #23 Dec 29 2018 21:03:57

%S 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,

%T 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,

%U 51,92,52,21,9,3,2,1,58,121,69,30,10,6,2,1,67,149

%N 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.

%H Alois P. Heinz, <a href="/A239223/b239223.txt">Rows n = 1..65, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Standard_deviation">Standard deviation</a>

%e Triangle T(n,k) begins:

%e 1;

%e 2;

%e 3;

%e 4, 1;

%e 6, 1;

%e 8, 2, 1;

%e 10, 4, 1;

%e 12, 7, 2, 1;

%e 15, 10, 4, 1;

%e 19, 14, 6, 2, 1;

%e 23, 21, 7, 4, 1;

%e 25, 32, 14, 3, 2, 1;

%p b:= proc(n, i, m, s, c) `if`(n=0, x^floor(sqrt(s/c-(m/c)^2)),

%p `if`(i=1, b(0$2, m+n, s+n, c+n), add(b(n-i*j, i-1,

%p m+i*j, s+i^2*j, c+j), j=0..n/i)))

%p end:

%p T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0$3)):

%p seq(T(n), n=1..18);

%t 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 *)

%Y Column k=0 gives A238616.

%Y Row sums give A000041.

%Y Maximal index in row n is A140106(n).

%Y Cf. A239228.

%K nonn,tabf

%O 1,2

%A _Alois P. Heinz_, Mar 12 2014