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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.

%I A239223 #23 Dec 29 2018 21:03:57
%S A239223 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 A239223 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 A239223 51,92,52,21,9,3,2,1,58,121,69,30,10,6,2,1,67,149
%N 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.
%H A239223 Alois P. Heinz, <a href="/A239223/b239223.txt">Rows n = 1..65, flattened</a>
%H A239223 Wikipedia, <a href="https://en.wikipedia.org/wiki/Standard_deviation">Standard deviation</a>
%e A239223 Triangle T(n,k) begins:
%e A239223    1;
%e A239223    2;
%e A239223    3;
%e A239223    4,  1;
%e A239223    6,  1;
%e A239223    8,  2,  1;
%e A239223   10,  4,  1;
%e A239223   12,  7,  2, 1;
%e A239223   15, 10,  4, 1;
%e A239223   19, 14,  6, 2, 1;
%e A239223   23, 21,  7, 4, 1;
%e A239223   25, 32, 14, 3, 2, 1;
%p A239223 b:= proc(n, i, m, s, c) `if`(n=0, x^floor(sqrt(s/c-(m/c)^2)),
%p A239223       `if`(i=1, b(0$2, m+n, s+n, c+n), add(b(n-i*j, i-1,
%p A239223        m+i*j, s+i^2*j, c+j), j=0..n/i)))
%p A239223     end:
%p A239223 T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0$3)):
%p A239223 seq(T(n), n=1..18);
%t A239223 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 A239223 Column k=0 gives A238616.
%Y A239223 Row sums give A000041.
%Y A239223 Maximal index in row n is A140106(n).
%Y A239223 Cf. A239228.
%K A239223 nonn,tabf
%O A239223 1,2
%A A239223 _Alois P. Heinz_, Mar 12 2014