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%I #64 May 16 2020 01:31:09
%S 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,
%T 349,418,543,648,827,992,1251,1495,1866,2230,2758,3289,4033,4803,5852,
%U 6949,8411,9973,12005,14194,17002,20060,23919,28153,33426,39256,46438
%N Number of 4's in the last section of the set of partitions of n.
%C Zero together with the first differences of A024788.
%C Also number of 4's in all partitions of n that do not contain 1 as a part.
%C 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
%C 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
%H Alois P. Heinz, <a href="/A182714/b182714.txt">Table of n, a(n) for n = 1..1000</a>
%F 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
%e 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.
%e From _Omar E. Pol_, Oct 25 2012: (Start)
%e --------------------------------------
%e Last section Number
%e of the set of of
%e partitions of 8 4's
%e --------------------------------------
%e 8 .............................. 0
%e 4 + 4 .......................... 2
%e 5 + 3 .......................... 0
%e 6 + 2 .......................... 0
%e 3 + 3 + 2 ...................... 0
%e 4 + 2 + 2 ...................... 1
%e 2 + 2 + 2 + 2 .................. 0
%e . 1 .......................... 0
%e . 1 ...................... 0
%e . 1 ...................... 0
%e . 1 .................. 0
%e . 1 ...................... 0
%e . 1 .................. 0
%e . 1 .................. 0
%e . 1 .............. 0
%e . 1 .................. 0
%e . 1 .............. 0
%e . 1 .............. 0
%e . 1 .......... 0
%e . 1 .......... 0
%e . 1 ...... 0
%e . 1 .. 0
%e ------------------------------------
%e . 6 - 3 = 3
%e .
%e 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).
%e (End)
%p b:= proc(n, i) option remember; local g, h;
%p if n=0 then [1, 0]
%p elif i<2 then [0, 0]
%p else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
%p [g[1]+h[1], g[2]+h[2]+`if`(i=4, h[1], 0)]
%p fi
%p end:
%p a:= n-> b(n, n)[2]:
%p seq (a(n), n=1..70); # _Alois P. Heinz_, Mar 19 2012
%t 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 *)
%t 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_ *)
%t Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 4], {n, 52}] (* _Robert Price_, May 15 2020 *)
%o (Sage) A182714 = lambda n: sum(list(p).count(4) for p in Partitions(n) if 1 not in p)
%Y Column 4 of A194812.
%Y Cf. A015739, A024788, A135010, A138121, A182703, A182712, A182713, A240058.
%K nonn
%O 1,8
%A _Omar E. Pol_, Nov 13 2011
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