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A255767 Triangle read by rows: T(n,k) = sum of all parts of all partitions of n into k distinct parts.

%I A255767 #30 Jun 07 2024 04:46:49
%S A255767 1,4,6,3,12,8,10,25,24,36,6,14,77,14,32,104,40,27,153,90,40,220,150,
%T A255767 10,22,297,275,22,72,348,444,60,26,481,676,130,56,616,938,280,60,660,
%U A255767 1455,450,15,80,880,1872,832,32,34,1003,2618,1309,85,108,1224,3312,2106,180,38,1349,4465,3078,380,120,1620,5540,4540,720
%N A255767 Triangle read by rows: T(n,k) = sum of all parts of all partitions of n into k distinct parts.
%C A255767 Row n has length A003056(n) hence the first element of column k is in row A000217(n).
%C A255767 The first element of column k is A000217(k).
%C A255767 Column 1 is A038040.
%H A255767 Alois P. Heinz, <a href="/A255767/b255767.txt">Rows n = 1..500, flattened</a>
%F A255767 T(n,k) = n * A116608(n,k).
%e A255767 Triangle begins:
%e A255767     1;
%e A255767     4;
%e A255767     6,    3;
%e A255767    12,    8;
%e A255767    10,   25;
%e A255767    24,   36,    6;
%e A255767    14,   77,   14;
%e A255767    32,  104,   40;
%e A255767    27,  153,   90;
%e A255767    40,  220,  150,   10;
%e A255767    22,  297,  275,   22;
%e A255767    72,  348,  444,   60;
%e A255767    26,  481,  676,  130;
%e A255767    56,  616,  938,  280;
%e A255767    60,  660, 1455,  450,  15;
%e A255767    80,  880, 1872,  832,  32;
%e A255767    34, 1003, 2618, 1309,  85;
%e A255767   108, 1224, 3312, 2106, 180;
%e A255767    38, 1349, 4465, 3078, 380;
%e A255767   ...
%p A255767 A255767 := proc(n,k)
%p A255767     n*A116608(n,k) ;
%p A255767 end proc:
%p A255767 for n from 1 to 20 do
%p A255767     for k from 1 to A003056(n) do
%p A255767         printf("%d,",A255767(n,k)) ;
%p A255767     end do:
%p A255767     printf("\n") ;
%p A255767 end do: # _R. J. Mathar_, Jul 10 2015
%p A255767 # second Maple program:
%p A255767 b:= proc(n, i) option remember; local j; if n=0 then 1
%p A255767       elif i<1 then 0 else []; for j from 0 to n/i do zip((x, y)
%p A255767       ->x+y, %, [`if`(j>0, 0, [][]), b(n-i*j, i-1)], 0) od; %[] fi
%p A255767     end:
%p A255767 T:= n-> subsop(1=NULL, n*[b(n, n)])[]:
%p A255767 seq(T(n), n=1..30);  # _Alois P. Heinz_, Jul 26 2015
%t A255767 nmax = 30; T = Rest[CoefficientList[#, t]]& /@ Rest[CoefficientList[-1 + Product[1 + t x^j/(1 - x^j), {j, 1, nmax}] + O[x]^(nmax+1), x]];
%t A255767 Table[n*T[[n]], {n, 1, nmax}] // Flatten (* _Jean-François Alcover_, May 19 2018 *)
%Y A255767 Cf. A000217, A003056, A038040, A066186 (row sums), A116608, A255768.
%K A255767 nonn,tabf,look
%O A255767 1,2
%A A255767 _Omar E. Pol_, May 21 2015
%E A255767 a(7) and beyond from _R. J. Mathar_, Jul 10 2015