cp's OEIS Frontend

A384999 Irregular triangle read by rows: T(n,k) is the total number of partitions of all numbers <= n with k designated summands, n >= 0, 0 <= k <= A003056(n).

%I A384999 #24 Aug 06 2025 14:47:38
%S A384999 1,1,1,1,4,1,8,1,1,15,4,1,21,13,1,33,28,1,1,41,58,4,1,56,103,13,1,69,
%T A384999 170,35,1,87,269,77,1,1,99,404,158,4,1,127,579,298,13,1,141,810,529,
%U A384999 35,1,165,1116,880,86,1,189,1470,1431,183,1,1,220,1935,2214,371,4,1,238,2475,3348,701,13
%N A384999 Irregular triangle read by rows: T(n,k) is the total number of partitions of all numbers <= n with k designated summands, n >= 0, 0 <= k <= A003056(n).
%C A384999 When part i has multiplicity j > 0 exactly one part i is "designated".
%C A384999 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 A384999 Column k gives the partial sums of the column k of A385001.
%C A384999 Columns converge to A210843 which is also the partial sums of A000716.
%H A384999 Alois P. Heinz, <a href="/A384999/b384999.txt">Rows n = 0..1000, flattened</a>
%e A384999 Triangle begins:
%e A384999 ---------------------------------------------
%e A384999    n\k:   0    1     2      3     4    5   6
%e A384999 ---------------------------------------------
%e A384999    0 |    1;
%e A384999    1 |    1,   1;
%e A384999    2 |    1,   4;
%e A384999    3 |    1,   8,    1;
%e A384999    4 |    1,  15,    4;
%e A384999    5 |    1,  21,   13;
%e A384999    6 |    1,  33,   28,     1;
%e A384999    7 |    1,  41,   58,     4;
%e A384999    8 |    1,  56,  103,    13;
%e A384999    9 |    1,  69,  170,    35;
%e A384999   10 |    1,  87,  269,    77,    1;
%e A384999   11 |    1,  99,  404,   158,    4;
%e A384999   12 |    1, 127,  579,   298,   13;
%e A384999   13 |    1, 141,  810,   529,   35;
%e A384999   14 |    1, 165, 1116,   880,   86;
%e A384999   15 |    1, 189, 1470,  1431,  183,   1;
%e A384999   16 |    1, 220, 1935,  2214,  371,   4;
%e A384999   17 |    1, 238, 2475,  3348,  701,  13;
%e A384999   18 |    1, 277, 3156,  4894, 1269,  35;
%e A384999   19 |    1, 297, 3921,  7036, 2187,  86;
%e A384999   20 |    1, 339, 4866,  9871, 3639, 194;
%e A384999   21 |    1, 371, 5906, 13629, 5872, 402,  1;
%e A384999   ...
%p A384999 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A384999       b(n, i-1)+add(expand(b(n-i*j, i-1)*j*x), j=1..n/i)))
%p A384999     end:
%p A384999 g:= proc(n) option remember; `if`(n<0, 0, g(n-1)+b(n$2)) end:
%p A384999 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n)):
%p A384999 seq(T(n), n=0..20);  # _Alois P. Heinz_, Jul 22 2025
%Y A384999 Columns: A000012 (k=0), A024916 (k=1).
%Y A384999 Cf. A000217, A000716, A002024, A003056, A077285, A195825, A210843, A211970, A385001.
%K A384999 nonn,look,tabf
%O A384999 0,5
%A A384999 _Omar E. Pol_, Jul 22 2025