A136343 - OEIS

%I #28 Sep 18 2018 16:10:58

%S 1,1,2,2,4,4,6,7,10,11,14,16,21,23,29,32,40,43,52,57,69,75,88,96,113,

%T 122,141,153,177,190,216,233,265,285,320,345,387,415,461,495,551,589,

%U 650,695,767,818,896,957,1048,1116,1214,1293,1407,1495,1620,1721,1864

%N Number of partitions of n such that each summand is greater than or equal to the sum of the next two summands.

%C This sequence was suggested by _Moshe Shmuel Newman_. The idea came to him while reading a paper of Lev Shneerson.

%C Number of partitions of 2n into exactly n positive Fibonacci numbers. a(8) = 10: 82111111, 55111111, 53311111, 53221111, 52222111, 33331111, 33322111, 33222211, 32222221, 22222222. - _Alois P. Heinz_, Sep 18 2018

%H Alois P. Heinz, <a href="/A136343/b136343.txt">Table of n, a(n) for n = 0..3500</a>

%F From _Alois P. Heinz_, Sep 18 2018: (Start)

%F a(n) = [x^(2n) y^n] 1/Product_{j>=2} (1-y*x^A000045(j)).

%F a(n) = A319394(2n,n). (End)

%e a(5) = 4 because 4 of the 7 partitions of 5 have the required property: {5} {4,1} {3,2} {3,1,1}. The other 3 partitions of 5: {2,2,1} {2,1,1,1} and {1,1,1,1,1} each have an element which is < the sum of next two.

%p b:= proc(n, i, j) option remember; `if`(n=0, 1,

%p       `if`(i<1, 0, b(n-i, min(n-i, i,

%p       `if`(j=0, i, j-i)), i) +b(n, i-1, j)))

%p     end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Jul 29 2017

%t b[n_, i_, j_]:=b[n, i, j]=If[n==0, 1, If[i<1, 0, b[n - i, Min[n - i, i, If[j==0, i, j - i]], i] + b[n, i - 1, j]]]; Table[b[n, n, 0], {n, 0, 60}] (* _Indranil Ghosh_, Aug 01 2017, after Maple code *)

%Y Cf. A000045, A319394.

%K nonn

%O 0,3

%A _David S. Newman_, May 11 2008

%E Conjectured g.f. removed and a(0), a(35)-a(56) added by _Alois P. Heinz_, Jul 29 2017