This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385001 #82 Aug 12 2025 20:32:51
%S A385001 1,0,1,0,3,0,4,1,0,7,3,0,6,9,0,12,15,1,0,8,30,3,0,15,45,9,0,13,67,22,
%T A385001 0,18,99,42,1,0,12,135,81,3,0,28,175,140,9,0,14,231,231,22,0,24,306,
%U A385001 351,51,0,24,354,551,97,1,0,31,465,783,188,3,0,18,540,1134,330,9
%N A385001 Irregular triangle read by rows: T(n,k) is the number of partitions of n with k designated summands, n >= 0, 0 <= k <= A003056(n).
%C A385001 The divisor function sigma_1(n) = A000203(n) is also the number of partitions of n with only one designated summand, n >= 1.
%C A385001 When part i has multiplicity j > 0 exactly one part i is "designated".
%C A385001 The length of the row n is A002024(n+1) = 1 + A003056(n), hence the first positive element in column k is in the row A000217(k).
%C A385001 Alternating row sums give A329157.
%C A385001 Columns converge to A000716.
%C A385001 This triangle equals A060043 with reversed rows and an additional column 0.
%H A385001 Alois P. Heinz, <a href="/A385001/b385001.txt">Rows n = 0..1000, flattened</a>
%F A385001 From _Alois P. Heinz_, Jul 18 2025: (Start)
%F A385001 Sum_{k>=1} k * T(n,k) = A293421(n).
%F A385001 T(A000096(n),n) = A000716(n). (End)
%F A385001 G.f.: Product_{i>0} 1 + (y*x^i)/(1 - x^i)^2. - _John Tyler Rascoe_, Jul 23 2025
%F A385001 Conjecture: For fixed k >= 1, Sum_{j=1..n} T(j,k) ~ Pi^(2*k) * n^(2*k) / ((2*k)! * (2*k+1)!). - _Vaclav Kotesovec_, Aug 01 2025
%e A385001 Triangle begins:
%e A385001 --------------------------------------------
%e A385001 n\k: 0 1 2 3 4 5 6
%e A385001 --------------------------------------------
%e A385001 0 | 1;
%e A385001 1 | 0, 1;
%e A385001 2 | 0, 3;
%e A385001 3 | 0, 4, 1;
%e A385001 4 | 0, 7, 3;
%e A385001 5 | 0, 6, 9;
%e A385001 6 | 0, 12, 15, 1;
%e A385001 7 | 0, 8, 30, 3;
%e A385001 8 | 0, 15, 45, 9;
%e A385001 9 | 0, 13, 67, 22;
%e A385001 10 | 0, 18, 99, 42, 1;
%e A385001 11 | 0, 12, 135, 81, 3;
%e A385001 12 | 0, 28, 175, 140, 9;
%e A385001 13 | 0, 14, 231, 231, 22;
%e A385001 14 | 0, 24, 306, 351, 51;
%e A385001 15 | 0, 24, 354, 551, 97, 1;
%e A385001 16 | 0, 31, 465, 783, 188, 3;
%e A385001 17 | 0, 18, 540, 1134, 330, 9;
%e A385001 18 | 0, 39, 681, 1546, 568, 22;
%e A385001 19 | 0, 20, 765, 2142, 918, 51;
%e A385001 20 | 0, 42, 945, 2835, 1452, 108;
%e A385001 21 | 0, 32, 1040, 3758, 2233, 208, 1;
%e A385001 ...
%e A385001 For n = 6 and k = 1 there are 12 partitions of 6 with only one designated summand as shown below:
%e A385001 6'
%e A385001 3'+ 3
%e A385001 3 + 3'
%e A385001 2'+ 2 + 2
%e A385001 2 + 2'+ 2
%e A385001 2 + 2 + 2'
%e A385001 1'+ 1 + 1 + 1 + 1 + 1
%e A385001 1 + 1'+ 1 + 1 + 1 + 1
%e A385001 1 + 1 + 1'+ 1 + 1 + 1
%e A385001 1 + 1 + 1 + 1'+ 1 + 1
%e A385001 1 + 1 + 1 + 1 + 1'+ 1
%e A385001 1 + 1 + 1 + 1 + 1 + 1'
%e A385001 So T(6,1) = 12, the same as A000203(6) = 12.
%e A385001 .
%e A385001 For n = 6 and k = 2 there are 15 partitions of 6 with two designated summands as shown below:
%e A385001 5'+ 1'
%e A385001 4'+ 2'
%e A385001 4'+ 1'+ 1
%e A385001 4'+ 1 + 1'
%e A385001 3'+ 1'+ 1 + 1
%e A385001 3'+ 1 + 1'+ 1
%e A385001 3'+ 1 + 1 + 1'
%e A385001 2'+ 2 + 1'+ 1
%e A385001 2'+ 2 + 1 + 1'
%e A385001 2 + 2'+ 1'+ 1
%e A385001 2 + 2'+ 1 + 1'
%e A385001 2'+ 1'+ 1 + 1 + 1
%e A385001 2'+ 1 + 1'+ 1 + 1
%e A385001 2'+ 1 + 1 + 1'+ 1
%e A385001 2'+ 1 + 1 + 1 + 1'
%e A385001 So T(6,2) = 15, the same as A002127(6) = 15.
%e A385001 .
%e A385001 For n = 6 and k = 3 there is only one partition of 6 with three designated summands as shown below:
%e A385001 3'+ 2'+ 1'
%e A385001 So T(6,3) = 1, the same as A002128(6) = 1.
%e A385001 There are 28 partitions of 6 with designated summands, so A077285(6) = 28.
%e A385001 .
%p A385001 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A385001 b(n, i-1)+add(expand(b(n-i*j, i-1)*j*x), j=1..n/i)))
%p A385001 end:
%p A385001 T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b(n$2)):
%p A385001 seq(T(n), n=0..20); # _Alois P. Heinz_, Jul 18 2025
%Y A385001 Columns: A000007 (k=0), A000203 (k=1), A002127 (k=2), A002128 (k=3), A365664 (k=4), A365665 (k=5), A384926 (k=6).
%Y A385001 Row sums give A077285.
%Y A385001 Cf. A000217, A002024, A003056, A060043, A196020, A252117, A329157, A365434, A384999.
%Y A385001 Cf. A000096, A000716, A116608, A293421, A384998, A384999.
%K A385001 nonn,look,tabf
%O A385001 0,5
%A A385001 _Omar E. Pol_, Jul 17 2025