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 A221529 Triangle read by rows: T(n,k) = A000203(k)*A000041(n-k). 6

%I

%S 1,1,3,2,3,4,3,6,4,7,5,9,8,7,6,7,15,12,14,6,12,11,21,20,21,12,12,8,15,

%T 33,28,35,18,24,8,15,22,45,44,49,30,36,16,15,13,30,66,60,77,42,60,24,

%U 30,13,18,42,90,88,105,66,84,40,45,26,18,12,56,126,120,154,90,132,56,75,39,36,12,28

%N Triangle read by rows: T(n,k) = A000203(k)*A000041(n-k).

%C Row sums give A066186.

%C Column 1 is A000041.

%C Leading diagonals 1-2: A000203, A000203.

%C T(n,k) is the number of partitions of n that contain k as a part multiplied by the sum of divisors of k.

%C It appears that T(n,k) is also the number of appearances of k in the last k section of the set of partitions of n multiplied by the sum of divisors of k.

%C T(n,k) is also the total number of parts in all partitions of k into equal parts multiplied by the number of ones in the j-th section of the set of partitions of n, where j = (n - k + 1).

%C Since A000203(k) has a symmetric representation then both T(n,k) and the partial sums of row n can be represented by symmetric polycubes - for more information see A237593 and A237270. For another version see A245099. - _Omar E. Pol_, Jul 15 2014

%F T(n,k) = sigma(k)*p(n-k) = A000203(k)*A027293(n,k).

%e For n = 6:

%e -------------------------

%e k A000203 T(6,k)

%e 1 1 * 7 = 7

%e 2 3 * 5 = 15

%e 3 4 * 3 = 12

%e 4 7 * 2 = 14

%e 5 6 * 1 = 6

%e 6 12 * 1 = 12

%e . A000041

%e -------------------------

%e So row 6 is [7, 15, 12, 14, 6, 12]. Note that the sum of row 6 is 7+15+12+14+6+12 = 66 equals A066186(6) = 6*p(6) = 6*11 = 66.

%e Triangle begins:

%e 1;

%e 1, 3;

%e 2, 3, 4;

%e 3, 6, 4, 7;

%e 5, 9, 8, 7, 6;

%e 7, 15, 12, 14, 6, 12;

%e 11, 21, 20, 21, 12, 12, 8;

%e 15, 33, 28, 35, 18, 24, 8, 15;

%e 22, 45, 44, 49, 30, 36, 16, 15, 13;

%e 30, 66, 60, 77, 42, 60, 24, 30, 13, 18;

%e 42, 90, 88, 105, 66, 84, 40, 45, 26, 18, 12;

%e 56, 126, 120, 154, 90, 132, 56, 75, 39, 36, 12, 28;

%o (PARI) T(n,k)=sigma(k)*numbpart(n-k) \\ _Charles R Greathouse IV_, Feb 19 2013

%Y Cf. A000041, A000203, A027293, A066186, A135010, A138137, A182703, A221530, A245095, A245099.

%K nonn,tabl

%O 1,3

%A _Omar E. Pol_, Jan 20 2013

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Last modified April 1 02:15 EDT 2020. Contains 333153 sequences. (Running on oeis4.)