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 A321449 #18 May 21 2021 04:16:24
%S A321449 1,0,1,0,1,2,0,1,2,3,0,1,4,5,5,0,1,4,8,8,7,0,1,6,13,19,16,11,0,1,6,17,
%T A321449 27,32,24,15,0,1,8,24,47,61,62,41,22,0,1,8,30,63,99,111,100,61,30,0,1,
%U A321449 10,38,94,158,209,210,170,95,42,0,1,10,45,119,229,328,382,348,259,136,56
%N A321449 Regular triangle read by rows where T(n,k) is the number of twice-partitions of n with a combined total of k parts.
%C A321449 A twice partition of n (A063834) is a choice of an integer partition of each part in an integer partition of n.
%H A321449 Alois P. Heinz, <a href="/A321449/b321449.txt">Rows n = 0..200, flattened</a>
%F A321449 O.g.f.: Product_{n >= 0} 1/(1 - x^n * (Sum_{0 <= k <= n} A008284(n,k) * t^k)).
%e A321449 Triangle begins:
%e A321449 1
%e A321449 0 1
%e A321449 0 1 2
%e A321449 0 1 2 3
%e A321449 0 1 4 5 5
%e A321449 0 1 4 8 8 7
%e A321449 0 1 6 13 19 16 11
%e A321449 0 1 6 17 27 32 24 15
%e A321449 0 1 8 24 47 61 62 41 22
%e A321449 0 1 8 30 63 99 111 100 61 30
%e A321449 The sixth row {0, 1, 6, 13, 19, 16, 11} counts the following twice-partitions:
%e A321449 (6) (33) (222) (2211) (21111) (111111)
%e A321449 (42) (321) (3111) (1111)(2) (111)(111)
%e A321449 (51) (411) (111)(3) (111)(21) (1111)(11)
%e A321449 (3)(3) (21)(3) (211)(2) (21)(111) (11111)(1)
%e A321449 (4)(2) (22)(2) (21)(21) (211)(11) (11)(11)(11)
%e A321449 (5)(1) (31)(2) (22)(11) (2111)(1) (111)(11)(1)
%e A321449 (3)(21) (221)(1) (11)(11)(2) (1111)(1)(1)
%e A321449 (32)(1) (3)(111) (111)(2)(1) (11)(11)(1)(1)
%e A321449 (4)(11) (31)(11) (11)(2)(11) (111)(1)(1)(1)
%e A321449 (41)(1) (311)(1) (2)(11)(11) (11)(1)(1)(1)(1)
%e A321449 (2)(2)(2) (11)(2)(2) (21)(11)(1) (1)(1)(1)(1)(1)(1)
%e A321449 (3)(2)(1) (2)(11)(2) (211)(1)(1)
%e A321449 (4)(1)(1) (21)(2)(1) (11)(2)(1)(1)
%e A321449 (2)(2)(11) (2)(11)(1)(1)
%e A321449 (22)(1)(1) (21)(1)(1)(1)
%e A321449 (3)(11)(1) (2)(1)(1)(1)(1)
%e A321449 (31)(1)(1)
%e A321449 (2)(2)(1)(1)
%e A321449 (3)(1)(1)(1)
%p A321449 g:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
%p A321449 g(n, i-1)+ `if`(i>n, 0, expand(g(n-i, i)*x)))
%p A321449 end:
%p A321449 b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
%p A321449 b(n, i-1)+ `if`(i>n, 0, expand(b(n-i, i)*g(i$2))))
%p A321449 end:
%p A321449 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A321449 seq(T(n), n=0..12); # _Alois P. Heinz_, Nov 11 2018
%t A321449 Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@ptn],Length[Join@@#]==k&],{ptn,IntegerPartitions[n]}]],{n,0,10},{k,0,n}]
%t A321449 (* Second program: *)
%t A321449 g[n_, i_] := g[n, i] = If[n == 0 || i == 1, x^n,
%t A321449 g[n, i - 1] + If[i > n, 0, Expand[g[n - i, i]*x]]];
%t A321449 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n,
%t A321449 b[n, i - 1] + If[i > n, 0, Expand[b[n - i, i]*g[i, i]]]];
%t A321449 T[n_] := CoefficientList[b[n, n], x];
%t A321449 T /@ Range[0, 12] // Flatten (* _Jean-François Alcover_, May 20 2021, after _Alois P. Heinz_ *)
%Y A321449 Row sums are A063834. Last column is A000041.
%Y A321449 Cf. A000219, A001970, A007716, A008284, A055884, A289501, A317449, A317532, A317533, A320801, A320808.
%K A321449 nonn,tabl
%O A321449 0,6
%A A321449 _Gus Wiseman_, Nov 10 2018