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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 Alois P. Heinz, Table of n, a(n) for n = 1..1000 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 (See A240058, ) - 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 *) 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. Cf. A015739, A024788, A135010, A138121, A182703, A182712, A182713, A240058. Sequence in context: A186545 A008623 A035546 * A262395 A198755 A134237 Adjacent sequences:  A182711 A182712 A182713 * A182715 A182716 A182717 KEYWORD nonn AUTHOR Omar E. Pol, Nov 13 2011 STATUS approved

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