cp's OEIS Frontend

A354468 Number of possible ordered pairs (n_1, S) where (n_1, n_2, ..., n_k) is a partition of n, n_1 is the largest element of the partition, and S = Sum_{j=1..k} n_j^2.

%I A354468 #32 Apr 24 2025 17:04:28
%S A354468 1,1,2,3,5,7,11,15,22,29,39,50,66,83,104,127,157,188,225,265,312,359,
%T A354468 418,479,547,620,700,786,884,987,1094,1214,1348,1479,1627,1779,1945,
%U A354468 2122,2313,2505,2719,2934,3161,3412,3666,3932,4218,4511,4820,5140,5477,5825
%N A354468 Number of possible ordered pairs (n_1, S) where (n_1, n_2, ..., n_k) is a partition of n, n_1 is the largest element of the partition, and S = Sum_{j=1..k} n_j^2.
%C A354468 In categorical data with a sample of size n distributed over at least 1 and at most n distinct categorical types, if a dataset is summarized by an ordered pair of two numbers -- the number of observations of the most frequent type and the sum of squares of the frequencies of all types -- then a(n) gives the number of distinguishable ordered pairs across all possible datasets.
%D A354468 Noah A. Rosenberg, Mathematical Properties of Population-Genetic Statistics, Princeton University Press, 2025, page 112.
%H A354468 Alois P. Heinz, <a href="/A354468/b354468.txt">Table of n, a(n) for n = 0..800</a>
%H A354468 Noah A. Rosenberg and Donna M. Zulman, <a href="https://doi.org/10.1111/1475-6773.13263">Measures of care fragmentation: mathematical insights from population genetics</a>, Health Services Research 55 (2020), 318-327.
%e A354468 For n=4 the a(4)=5 ordered pairs are (4,16), (3,10), (2,8), (2,6), and (1,4).
%p A354468 b:= proc(n, i) option remember; `if`(n=0 or i=1, {n},
%p A354468      {b(n, i-1)[], map(x-> x+i^2, b(n-i, min(n-i, i)))[]})
%p A354468     end:
%p A354468 a:= n-> add(nops(b(n-i, min(n-i, i))), i=signum(n)..n):
%p A354468 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jun 02 2022
%t A354468 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {n},
%t A354468      Union@Flatten@{b[n, i-1], #+i^2& /@ b[n-i, Min[n-i, i]]}];
%t A354468 a[n_] := Sum[Length[b[n-i, Min[n-i, i]]], {i, Sign[n], n}];
%t A354468 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Jun 05 2022, after _Alois P. Heinz_ *)
%Y A354468 Bounded below by A069999. Bounded above by A000041 and by A000125(n-1).
%Y A354468 Cf. A354800.
%K A354468 nonn
%O A354468 0,3
%A A354468 _Noah A Rosenberg_, Jun 02 2022
%E A354468 a(16)-a(51) from _Alois P. Heinz_, Jun 02 2022