

A360673


Number of multisets of positive integers whose right half (exclusive) sums to n.


17



1, 2, 7, 13, 27, 37, 73, 89, 156, 205, 315, 387, 644, 749, 1104, 1442, 2015, 2453, 3529, 4239, 5926, 7360, 9624, 11842, 16115, 19445, 25084, 31137, 39911, 48374, 62559, 75135, 95263, 115763, 143749, 174874, 218614, 261419, 321991, 388712, 477439, 569968, 698493
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OFFSET

0,2


LINKS



FORMULA

G.f.: 1 + Sum_{k>=1} x^k*(2  x^k)/((1  x^k)^(k+2) * Product_{j=1..k1} (1x^j)).  Andrew Howroyd, Mar 11 2023


EXAMPLE

The a(0) = 1 through a(3) = 13 multisets:
{} {1,1} {1,2} {1,3}
{1,1,1} {2,2} {2,3}
{1,1,2} {3,3}
{1,2,2} {1,1,3}
{2,2,2} {1,2,3}
{1,1,1,1} {1,3,3}
{1,1,1,1,1} {2,2,3}
{2,3,3}
{3,3,3}
{1,1,1,2}
{1,1,1,1,2}
{1,1,1,1,1,1}
{1,1,1,1,1,1,1}
For example, the multiset y = {1,1,1,1,2} has right half (exclusive) {1,2}, with sum 3, so y is counted under a(3).


MATHEMATICA

Table[Length[Select[Join@@IntegerPartitions/@Range[0, 3*k], Total[Take[#, Floor[Length[#]/2]]]==k&]], {k, 0, 15}]


PROG

(PARI) seq(n)={my(s=1 + O(x*x^n), p=s); for(k=1, n, s += p*x^k*(2x^k)/(1x^k + O(x*x^(nk)))^(k+2); p /= 1  x^k); Vec(s)} \\ Andrew Howroyd, Mar 11 2023


CROSSREFS

First for prime indices, second for partitions, third for prime factors:


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



