

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).


4



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
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OFFSET

1,8


COMMENTS

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


LINKS

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


FORMULA

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


EXAMPLE

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;


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(ni*j, i1)], 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 (* JeanFrançois Alcover, Jan 08 2014 *)


CROSSREFS

Cf. A001935, A008289, A117148, A209318.
Sequence in context: A187451 A134542 A106254 * A111007 A176353 A103691
Adjacent sequences: A117144 A117145 A117146 * A117148 A117149 A117150


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Mar 07 2006


STATUS

approved



