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A221530 Triangle read by rows: T(n,k) = A000005(k)*A000041(n-k).

%I A221530 #47 Sep 04 2023 06:07:12
%S A221530 1,1,2,2,2,2,3,4,2,3,5,6,4,3,2,7,10,6,6,2,4,11,14,10,9,4,4,2,15,22,14,
%T A221530 15,6,8,2,4,22,30,22,21,10,12,4,4,3,30,44,30,33,14,20,6,8,3,4,42,60,
%U A221530 44,45,22,28,10,12,6,4,2,56,84,60,66,30,44,14,20,9,8,2,6
%N A221530 Triangle read by rows: T(n,k) = A000005(k)*A000041(n-k).
%C A221530 T(n,k) is the number of partitions of n that contain k as a part multiplied by the number of divisors of k.
%C A221530 It appears that T(n,k) is also the total number of appearances of k in the last k sections of the set of partitions of n multiplied by the number of divisors of k.
%C A221530 T(n,k) is also the number of 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 A221530 For another version see A245095. - _Omar E. Pol_, Jul 15 2014
%H A221530 Paolo Xausa, <a href="/A221530/b221530.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of the triangle, flattened)
%F A221530 T(n,k) = d(k)*p(n-k) = A000005(k)*A027293(n,k).
%e A221530 For n = 6:
%e A221530   -------------------------
%e A221530   k   A000005        T(6,k)
%e A221530   1      1  *  7   =    7
%e A221530   2      2  *  5   =   10
%e A221530   3      2  *  3   =    6
%e A221530   4      3  *  2   =    6
%e A221530   5      2  *  1   =    2
%e A221530   6      4  *  1   =    4
%e A221530   .         A000041
%e A221530   -------------------------
%e A221530 So row 6 is [7, 10, 6, 6, 4, 2]. Note that the sum of row 6 is 7+10+6+6+2+4 = 35 equals A006128(6).
%e A221530 .
%e A221530 Triangle begins:
%e A221530   1;
%e A221530   1,   2;
%e A221530   2,   2,  2;
%e A221530   3,   4,  2,  3;
%e A221530   5,   6,  4,  3,  2;
%e A221530   7,  10,  6,  6,  2,  4;
%e A221530   11, 14, 10,  9,  4,  4,  2;
%e A221530   15, 22, 14, 15,  6,  8,  2,  4;
%e A221530   22, 30, 22, 21, 10, 12,  4,  4,  3;
%e A221530   30, 44, 30, 33, 14, 20,  6,  8,  3,  4;
%e A221530   42, 60, 44, 45, 22, 28, 10, 12,  6,  4,  2;
%e A221530   56, 84, 60, 66, 30, 44, 14, 20,  9,  8,  2,  6;
%e A221530   ...
%t A221530 A221530row[n_]:=DivisorSigma[0,Range[n]]PartitionsP[n-Range[n]];Array[A221530row,10] (* _Paolo Xausa_, Sep 04 2023 *)
%o A221530 (PARI) row(n) = vector(n, i, numdiv(i)*numbpart(n-i)); \\ _Michel Marcus_, Jul 18 2014
%Y A221530 Similar to A221529.
%Y A221530 Columns 1-2: A000041, A139582. Leading diagonals 1-3: A000005, A000005, A062011. Row sums give A006128.
%Y A221530 Cf. A027293, A135010, A138137, A182703, A245095, A245099.
%K A221530 nonn,tabl
%O A221530 1,3
%A A221530 _Omar E. Pol_, Jan 19 2013