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A243746 Number of compositions of n^2 with exactly n occurrences of the largest part.

%I A243746 #10 Mar 04 2022 11:13:31
%S A243746 1,1,1,21,686,108598,134190162,581266801787,7792898359869376,
%T A243746 343349252968004533986,60917528224825622999788393,
%U A243746 57691110936849283646013592507915,280564704602761525363382338982479319450,5619591974217690324311922622790661532819536973
%N A243746 Number of compositions of n^2 with exactly n occurrences of the largest part.
%H A243746 Alois P. Heinz, <a href="/A243746/b243746.txt">Table of n, a(n) for n = 0..33</a>
%F A243746 a(n) = A238341(n^2,n).
%e A243746 a(3) = 21: 333, 111222, 112122, 112212, 112221, 121122, 121212, 121221, 122112, 122121, 122211, 211122, 211212, 211221, 212112, 212121, 212211, 221112, 221121, 221211, 222111.
%p A243746 b:= proc(n, p, i) option remember; `if`(n=0, p!,
%p A243746       `if`(i<1, 0, add(b(n-i*j, p+j, i-1)/j!, j=0..n/i)))
%p A243746     end:
%p A243746 a:= n-> add(b(n^2-i*n, n, i-1)/n!, i=0..n):
%p A243746 seq(a(n), n=0..15);
%t A243746 b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!,
%t A243746      If[i < 1, 0, Sum[b[n - i*j, p + j, i - 1]/j!, {j, 0, n/i}]]];
%t A243746 a[n_] := Sum[b[n^2 - i*n, n, i - 1]/n!, {i, 0, n}];
%t A243746 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Mar 04 2022, after _Alois P. Heinz_ *)
%Y A243746 Cf. A238341.
%K A243746 nonn
%O A243746 0,4
%A A243746 _Alois P. Heinz_, Jun 09 2014