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 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
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified April 20 00:03 EDT 2024. Contains 371798 sequences. (Running on oeis4.)