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A299769 Triangle read by rows: T(n,k) is the sum of all squares of the parts k in the last section of the set of partitions of n, with n >= 1, 1 <= k <= n.

%I A299769 #36 Dec 10 2019 12:10:57
%S A299769 1,1,4,2,0,9,3,8,0,16,5,4,9,0,25,7,16,18,16,0,36,11,12,18,16,25,0,49,
%T A299769 15,32,27,48,25,36,0,64,22,28,54,32,50,36,49,0,81,30,60,54,80,75,72,
%U A299769 49,64,0,100,42,60,90,80,100,72,98,64,81,0,121,56,108,126,160,125,180,98,128,81,100,0,144
%N A299769 Triangle read by rows: T(n,k) is the sum of all squares of the parts k in the last section of the set of partitions of n, with n >= 1, 1 <= k <= n.
%C A299769 The partial sums of the k-th column of this triangle give the k-th column of triangle A299768.
%C A299769 Note that the last section of the set of partitions of n is also the n-th section of the set of partitions of any positive integer >= n.
%H A299769 Alois P. Heinz, <a href="/A299769/b299769.txt">Rows n = 1..200, flattened</a>
%F A299769 T(n,k) = A299768(n,k) - A299768(n-1,k). - _Alois P. Heinz_, Jul 23 2018
%e A299769 Triangle begins:
%e A299769    1;
%e A299769    1,   4;
%e A299769    2,   0,   9;
%e A299769    3,   8,   0,  16;
%e A299769    5,   4,   9,   0,  25;
%e A299769    7,  16,  18,  16,   0,  36;
%e A299769   11,  12,  18,  16,  25,   0,  49;
%e A299769   15,  32,  27,  48,  25,  36,   0,  64;
%e A299769   22,  28,  54,  32,  50,  36,  49,   0,  81;
%e A299769   30,  60,  54,  80,  75,  72,  49,  64,   0, 100;
%e A299769   42,  60,  90,  80, 100,  72,  98,  64,  81,   0, 121;
%e A299769   56, 108, 126, 160, 125, 180,  98, 128,  81, 100,   0, 144;
%e A299769   ...
%e A299769 Illustration for the 4th row of triangle:
%e A299769 .
%e A299769 .                                  Last section of the set
%e A299769 .        Partitions of 4.          of the partitions of 4.
%e A299769 .       _ _ _ _                              _
%e A299769 .      |_| | | |  [1,1,1,1]                 | |  [1]
%e A299769 .      |_ _| | |  [2,1,1]                   | |  [1]
%e A299769 .      |_ _ _| |  [3,1]                _ _ _| |  [1]
%e A299769 .      |_ _|   |  [2,2]               |_ _|   |  [2,2]
%e A299769 .      |_ _ _ _|  [4]                 |_ _ _ _|  [4]
%e A299769 .
%e A299769 For n = 4 the last section of the set of partitions of 4 is [4], [2, 2], [1], [1], [1], so the squares of the parts are respectively [16], [4, 4], [1], [1], [1]. The sum of the squares of the parts 1 is 1 + 1 + 1 = 3. The sum of the squares of the parts 2 is 4 + 4 = 8. The sum of the squares of the parts 3 is 0 because there are no parts 3. The sum of the squares of the parts 4 is 16. So the fourth row of triangle is [3, 8, 0, 16].
%p A299769 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1+n*x, b(n, i-1)+
%p A299769       (p-> p+(coeff(p, x, 0)*i^2)*x^i)(b(n-i, min(n-i, i))))
%p A299769     end:
%p A299769 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)-b(n-1$2)):
%p A299769 seq(T(n), n=1..14);  # _Alois P. Heinz_, Jul 23 2018
%t A299769 b[n_, i_] := b[n, i] = If[n==0 || i==1, 1 + n*x, b[n, i-1] + Function[p, p + (Coefficient[p, x, 0]*i^2)*x^i][b[n-i, Min[n-i, i]]]];
%t A299769 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, n] - b[n-1, n-1]];
%t A299769 T /@ Range[14] // Flatten (* _Jean-François Alcover_, Dec 10 2019, after _Alois P.heinz_ *)
%Y A299769 Column 1 is A000041.
%Y A299769 Leading diagonal gives A000290, n >= 1.
%Y A299769 Second diagonal gives A000007.
%Y A299769 Row sums give A206440.
%Y A299769 Cf. A066183, A135010, A206438, A299768.
%K A299769 nonn,tabl
%O A299769 1,3
%A A299769 _Omar E. Pol_, Mar 20 2018