A117147 Triangle read by rows: T(n,k) is the number of partitions of n with k parts in which no part occurs more than 3 times (n>=1, k>=1).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 3, 4, 3, 1, 1, 4, 5, 4, 2, 1, 4, 7, 6, 3, 1, 1, 5, 8, 9, 5, 1, 1, 5, 10, 11, 8, 3, 1, 6, 12, 14, 11, 5, 1, 1, 6, 14, 18, 15, 8, 2, 1, 7, 16, 23, 20, 11, 4, 1, 7, 19, 27, 27, 17, 6, 1, 1, 8, 21, 33, 34, 23, 10, 2, 1, 8, 24, 39, 43, 32, 15, 4, 1, 9

Offset: 1

Author

Emeric Deutsch, Mar 07 2006

Keywords

• nonn
• tabf

Comments

Row n has floor(sqrt(6n+6)-3/2) terms. Row sums yield A001935. Sum(k*T(n,k),k>=0) = A117148(n).

Examples

			T(7,3) = 4 because we have [5,1,1], [4,2,1], [3,3,1] and [3,2,2].
Triangle starts:
1;
1, 1;
1, 1, 1;
1, 2, 1;
1, 2, 2, 1;
1, 3, 3, 2;
1, 3, 4, 3, 1;
		

Links

• Alois P. Heinz, Rows n = 1..350, flattened

Crossrefs

Cf. A001935, A008289, A117148, A209318.

Programs

Maple

g:=-1+product(1+t*x^j+t^2*x^(2*j)+t^3*x^(3*j),j=1..35): gser:=simplify(series(g,x=0,23)): for n from 1 to 18 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 18 do seq(coeff(P[n],t^j),j=1..floor(sqrt(6*n+6)-3/2)) od; # yields sequence in triangular form
# 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 min(3, n/i) do
      zip((x, y)->x+y, %, [0$j, b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> subsop(1=NULL, [b(n, n)])[]:
seq(T(n), n=1..20);  # Alois P. Heinz, Jan 08 2013

Mathematica

max = 18; g = -1+Product[1+t*x^j+t^2*x^(2j)+t^3*x^(3j), {j, 1, max}]; t[n_, k_] := SeriesCoefficient[g, {x, 0, n}, {t, 0, k}]; Table[DeleteCases[Table[t[n, k], {k, 1, n}], 0], {n, 1, max}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

Formula

G.f.: G(t,x) = -1+product(1+tx^j+t^2*x^(2j)+t^3*x^(3j), j=1..infinity).