

A136343


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


2



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, 122, 141, 153, 177, 190, 216, 233, 265, 285, 320, 345, 387, 415, 461, 495, 551, 589, 650, 695, 767, 818, 896, 957, 1048, 1116, 1214, 1293, 1407, 1495, 1620, 1721, 1864
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OFFSET

0,3


COMMENTS

This sequence was suggested by Moshe Shmuel Newman. The idea came to him while reading a paper of Lev Shneerson.
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


LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..3500


FORMULA

From Alois P. Heinz, Sep 18 2018: (Start)
a(n) = [x^(2n) y^n] 1/Product_{j>=2} (1y*x^A000045(j)).
a(n) = A319394(2n,n). (End)


EXAMPLE

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.


MAPLE

b:= proc(n, i, j) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(ni, min(ni, i,
`if`(j=0, i, ji)), i) +b(n, i1, j)))
end:
a:= n> b(n$2, 0):
seq(a(n), n=0..60); # Alois P. Heinz, Jul 29 2017


MATHEMATICA

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 *)


CROSSREFS

Cf. A000045, A319394.
Sequence in context: A319403 A029008 A240844 * A161254 A241313 A241317
Adjacent sequences: A136340 A136341 A136342 * A136344 A136345 A136346


KEYWORD

nonn


AUTHOR

David S. Newman, May 11 2008


EXTENSIONS

Conjectured g.f. removed and a(0), a(35)a(56) added by Alois P. Heinz, Jul 29 2017


STATUS

approved



