A083690 - OEIS

A083690

a(n) = number of partitions of n wherein the sum of the 1's is no more than the sum of the other parts.

0, 1, 2, 4, 5, 9, 12, 19, 25, 37, 49, 70, 90, 124, 161, 216, 275, 363, 460, 597, 750, 960, 1199, 1519, 1881, 2359, 2909, 3617, 4430, 5469, 6666, 8173, 9912, 12079, 14586, 17680, 21252, 25630, 30695, 36848, 43956, 52547, 62469, 74383, 88132, 104556

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OFFSET

1,3

LINKS

Table of n, a(n) for n=1..46.

FORMULA

a(n) = A000041(n)-A000041(floor((n-1)/2)). - Vladeta Jovovic, Jun 15 2003

EXAMPLE

a(5)=5 because 5 = 1+4 = 1+1+3 = 1+2+2 = 2+3. (1+1+1+2 and 1+1+1+1+1 have too many ones.)

MATHEMATICA

<< DiscreteMath`Combinatorica`; f[n_] := Block[{c = 0, k = 1, p = Partitions[n], l = PartitionsP[n]}, While[k < l, c1 = Count[p[[k]], 1]; If[ Plus @@ Take[ p[[k]], Length[ p[[k]]] - c1] >= c1, c++ ]; k++ ]; c]; Table[ f[n], {n, 1, 25}]

PROG

(PARI) mybinary(n, m)=local(u); u=binary(n); concat(vector(m-length(u), i, 0), u); { for (n=2, 16, y=vector(2^(n-1)); for (j=0, 2^(n-1)-1, x=concat(mybinary(j, n-1), [1]); y[j+1]=vector(n); c=0; for (k=1, n, if (x[k]==1, y[j+1][k]=c+1; c=0, c++)); y[j+1]=vecsort(y[j+1])); y=vecsort(y, QQ=QQ, 2);

for (i=1, 2^(n-1)-1, z=1; if (type(y[i])=="t_VEC", while (y[i]==y[i+z], y[i+z]=0; z++))); sc=0; for(i=1, 2^(n-1), if (type(y[i])=="t_VEC", os=0; rs=0; for (q=1, n, if (y[i][q]==1, os++, rs+=y[i][q])); if (os<=rs, sc++))); print1(sc", ") )}

CROSSREFS

Sequence in context: A241734 A371171 A039898 * A241824 A144121 A240576

Adjacent sequences: A083687 A083688 A083689 * A083691 A083692 A083693

KEYWORD

nonn

AUTHOR

Jon Perry, Jun 15 2003

EXTENSIONS

More terms from Vladeta Jovovic and Don Reble, Jun 15 2003

STATUS

approved