A216669
Total number of parts in all partitions of n into 2 sizes of parts (A002133).
Original entry on oeis.org
2, 5, 14, 20, 39, 52, 74, 102, 134, 158, 208, 259, 284, 361, 409, 488, 538, 634, 678, 838, 857, 1006, 1038, 1270, 1264, 1495, 1500, 1776, 1761, 2084, 2062, 2443, 2300, 2795, 2680, 3194, 3076, 3544, 3403, 4080, 3804, 4518, 4282, 5037, 4673, 5626, 5127, 6088
Offset: 3
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nn=40;ss=Sum[Sum[y^2 x^(i+j)/(1-y x^i)/(1-y x^j),{j,1,i-1}], {i,1,nn}]; CoefficientList[Series[D[ss,y]/.y->1,{x,0,nn}],x]
A255767
Triangle read by rows: T(n,k) = sum of all parts of all partitions of n into k distinct parts.
Original entry on oeis.org
1, 4, 6, 3, 12, 8, 10, 25, 24, 36, 6, 14, 77, 14, 32, 104, 40, 27, 153, 90, 40, 220, 150, 10, 22, 297, 275, 22, 72, 348, 444, 60, 26, 481, 676, 130, 56, 616, 938, 280, 60, 660, 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
Offset: 1
Triangle begins:
1;
4;
6, 3;
12, 8;
10, 25;
24, 36, 6;
14, 77, 14;
32, 104, 40;
27, 153, 90;
40, 220, 150, 10;
22, 297, 275, 22;
72, 348, 444, 60;
26, 481, 676, 130;
56, 616, 938, 280;
60, 660, 1455, 450, 15;
80, 880, 1872, 832, 32;
34, 1003, 2618, 1309, 85;
108, 1224, 3312, 2106, 180;
38, 1349, 4465, 3078, 380;
...
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A255767 := proc(n,k)
n*A116608(n,k) ;
end proc:
for n from 1 to 20 do
for k from 1 to A003056(n) do
printf("%d,",A255767(n,k)) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, Jul 10 2015
# second Maple program:
b:= proc(n, i) option remember; local j; if n=0 then 1
elif i<1 then 0 else []; for j from 0 to n/i do zip((x, y)
->x+y, %, [`if`(j>0, 0, [][]), b(n-i*j, i-1)], 0) od; %[] fi
end:
T:= n-> subsop(1=NULL, n*[b(n, n)])[]:
seq(T(n), n=1..30); # Alois P. Heinz, Jul 26 2015
-
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]];
Table[n*T[[n]], {n, 1, nmax}] // Flatten (* Jean-François Alcover, May 19 2018 *)
Showing 1-2 of 2 results.
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