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A229468 Number T(n,k) of parts of each size k^2 in all partitions of n^2 into squares; triangle T(n,k), 1 <= k <= n, read by rows.

%I A229468 #27 Oct 27 2023 22:06:58
%S A229468 1,4,1,15,3,1,50,11,2,1,156,35,10,4,1,460,101,36,14,4,1,1296,298,105,
%T A229468 44,16,6,1,3522,798,300,130,56,23,6,1,9255,2154,827,377,174,82,31,9,1,
%U A229468 23672,5490,2164,1015,502,243,108,43,10,1,59050,13914,5525,2658,1350,705,343,154,55,13,1
%N A229468 Number T(n,k) of parts of each size k^2 in all partitions of n^2 into squares; triangle T(n,k), 1 <= k <= n, read by rows.
%H A229468 Alois P. Heinz, <a href="/A229468/b229468.txt">Rows n = 1..141, flattened</a> (Rows n = 1..21 from Christopher Hunt Gribble)
%H A229468 Christopher Hunt Gribble, <a href="/A229468/a229468.cpp.txt">C++ program</a>
%F A229468 Sum_{k=1..n} T(n,k) * k^2 = A037444(n) * n^2.
%e A229468 For n = 3, the 4 partitions are:
%e A229468 Square side 1 2 3
%e A229468             9 0 0
%e A229468             5 1 0
%e A229468             1 2 0
%e A229468             0 0 1
%e A229468 Total      15 3 1
%e A229468 So T(3,1) = 15, T(3,2) = 3, T(3,3) = 1.
%e A229468 The triangle begins:
%e A229468 .\ k    1     2     3     4     5     6     7     8     9 ...
%e A229468 .n
%e A229468 .1      1
%e A229468 .2      4     1
%e A229468 .3     15     3     1
%e A229468 .4     50    11     2     1
%e A229468 .5    156    35    10     4     1
%e A229468 .6    460   101    36    14     4     1
%e A229468 .7   1296   298   105    44    16     6     1
%e A229468 .8   3522   798   300   130    56    23     6     1
%e A229468 .9   9255  2154   827   377   174    82    31     9     1
%e A229468 10  23672  5490  2164  1015   502   243   108    43    10 ...
%e A229468 11  59050 13914  5525  2658  1350   705   343   154    55 ...
%p A229468 b:= proc(n, i) option remember;
%p A229468       `if`(n=0 or i=1, 1+n*x, b(n, i-1)+
%p A229468       `if`(i^2>n, 0, (g->g+coeff(g, x, 0)*x^i)(b(n-i^2, i))))
%p A229468     end:
%p A229468 T:= n-> (p->seq(coeff(p, x, i), i=1..n))(b(n^2, n)):
%p A229468 seq(T(n), n=1..14);  # _Alois P. Heinz_, Sep 24 2013
%t A229468 b[n_, i_] := b[n, i] = If[n==0 || i==1, 1+n*x, b[n, i-1] + If[i^2>n, 0, Function[ {g}, g+Coefficient[g, x, 0]*x^i][b[n-i^2, i]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 1, n}]][ b[n^2, n]]; Table[T[n], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Mar 09 2015, after _Alois P. Heinz_ *)
%Y A229468 Row sums give: A229239.
%Y A229468 Cf. A037444.
%K A229468 nonn,tabl
%O A229468 1,2
%A A229468 _Christopher Hunt Gribble_, Sep 24 2013