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%I #47 Jun 07 2018 21:58:44
%S 1,2,4,4,4,9,7,12,9,16,12,16,18,16,25,19,32,36,32,25,36,30,44,54,48,
%T 50,36,49,45,76,81,96,75,72,49,64,67,104,135,128,125,108,98,64,81,97,
%U 164,189,208,200,180,147,128,81,100,139,224,279,288,300,252,245,192,162,100,121
%N Triangle read by rows: T(n,k) = sum of all squares of the parts k in all partitions of n, with n >= 1, 1 <= k <= n.
%H Alois P. Heinz, <a href="/A299768/b299768.txt">Rows n = 1..200, flattened</a>
%F T(n,k) = (k^2)*A066633(n,k) = k*A138785(n,k). - _Omar E. Pol_, Jun 07 2018
%e Triangle begins:
%e 1;
%e 2, 4;
%e 4, 4, 9;
%e 7, 12, 9, 16;
%e 12, 16, 18, 16, 25,
%e 19, 32, 36, 32, 25, 36;
%e 30, 44, 54, 48, 50, 36, 49;
%e ...
%e For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1], so the squares of the parts are respectively [16], [4, 4], [9, 1], [4, 1, 1], [1, 1, 1, 1]. The sum of the squares of the parts 1 is 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7. The sum of the squares of the parts 2 is 4 + 4 + 4 = 12. The sum of the squares of the parts 3 is 9. The sum of the squares of the parts 4 is 16. So the fourth row of triangle is [7, 12, 9, 16].
%p b:= proc(n, i) option remember; `if`(n=0 or i=1, 1+n*x, b(n, i-1)+
%p (p-> p+(coeff(p, x, 0)*i^2)*x^i)(b(n-i, min(n-i, i))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
%p seq(T(n), n=1..14); # _Alois P. Heinz_, Mar 20 2018
%t b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1 + n*x, b[n, i - 1] + # + (Coefficient[#, x, 0]*i^2*x^i)&[b[n - i, Min[n - i, i]]]];
%t T[n_] := Table[Coefficient[#, x, i], {i, 1, n}]&[b[n, n]];
%t Table[T[n], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, May 22 2018, after _Alois P. Heinz_ *)
%o (PARI) row(n) = {v = vector(n); forpart(p=n, for(k=1, #p, v[p[k]] += p[k]^2;);); v;} \\ _Michel Marcus_, Mar 20 2018
%Y Column 1 is A000070.
%Y Leading diagonal is A000290, n >= 1.
%Y Row sums give A066183.
%Y Both A180681 and A206561 have the same row sums as this triangle.
%Y Cf. A066633, A138785.
%K nonn,tabl
%O 1,2
%A _Omar E. Pol_, Mar 19 2018
%E More terms from _Michel Marcus_, Mar 20 2018
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