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A182714 Number of 4's in the last section of the set of partitions of n. 22
0, 0, 0, 1, 0, 1, 1, 3, 2, 5, 5, 10, 10, 17, 19, 31, 34, 51, 60, 86, 100, 139, 165, 223, 265, 349, 418, 543, 648, 827, 992, 1251, 1495, 1866, 2230, 2758, 3289, 4033, 4803, 5852, 6949, 8411, 9973, 12005, 14194, 17002, 20060, 23919, 28153, 33426, 39256, 46438 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
Zero together with the first differences of A024788.
Also number of 4's in all partitions of n that do not contain 1 as a part.
a(n) is the number of partitions of n such that m(1) < m(3), where m = multiplicity; e.g., a(7) counts these 3 partitions: [4, 3], [3, 3, 1], [3, 2, 2]. - Clark Kimberling, Apr 01 2014
The last section of the set of partitions of n is also the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Apr 07 2014
LINKS
FORMULA
It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3) + a(n+4), n >= 0. - Omar E. Pol, Feb 04 2012
EXAMPLE
a(8) = 3 counts the 4's in 8 = 4+4 = 4+2+2. The 4's in 8 = 4+3+1 = 4+2+1+1 = 4+1+1+1+1 do not count.
From Omar E. Pol, Oct 25 2012: (Start)
--------------------------------------
Last section Number
of the set of of
partitions of 8 4's
--------------------------------------
8 .............................. 0
4 + 4 .......................... 2
5 + 3 .......................... 0
6 + 2 .......................... 0
3 + 3 + 2 ...................... 0
4 + 2 + 2 ...................... 1
2 + 2 + 2 + 2 .................. 0
. 1 .......................... 0
. 1 ...................... 0
. 1 ...................... 0
. 1 .................. 0
. 1 ...................... 0
. 1 .................. 0
. 1 .................. 0
. 1 .............. 0
. 1 .................. 0
. 1 .............. 0
. 1 .............. 0
. 1 .......... 0
. 1 .......... 0
. 1 ...... 0
. 1 .. 0
------------------------------------
. 6 - 3 = 3
.
In the last section of the set of partitions of 8 the difference between the sum of the fourth column and the sum of the fifth column is 6 - 3 = 3 equaling the number of 4's, so a(8) = 3 (see also A024788).
(End)
MAPLE
b:= proc(n, i) option remember; local g, h;
if n=0 then [1, 0]
elif i<2 then [0, 0]
else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
[g[1]+h[1], g[2]+h[2]+`if`(i=4, h[1], 0)]
fi
end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..70); # Alois P. Heinz, Mar 19 2012
MATHEMATICA
z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 3]], {n, 0, z}] (* Clark Kimberling, Apr 01 2014 *)
b[n_, i_] := b[n, i] = Module[{g, h}, If[n==0, {1, 0}, If[i<2, {0, 0}, g = b[n, i-1]; h = If[i>n, {0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + If[i==4, h[[1]], 0]}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *)
Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 4], {n, 52}] (* Robert Price, May 15 2020 *)
PROG
(Sage) A182714 = lambda n: sum(list(p).count(4) for p in Partitions(n) if 1 not in p)
CROSSREFS
Column 4 of A194812.
Sequence in context: A008623 A035546 A339406 * A343342 A338470 A262395
KEYWORD
nonn
AUTHOR
Omar E. Pol, Nov 13 2011
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)