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 A239873 Number of strict partitions of 2n + 1 having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even. 4
 0, 0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 43, 51, 61, 74, 91, 113, 144, 184, 239, 311, 407, 530, 692, 895, 1155, 1478, 1882, 2375, 2983, 3715, 4602, 5660, 6925, 8418, 10187, 12257, 14686, 17514, 20809, 24624, 29049, 34154, 40051, 46842, 54668, 63667 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Let c(n) be the number of strict partitions (that is, every part has multiplicity 1) of 2n having 1 more even part than odd, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are even.  This sequence is nondecreasing, unlike A239871, of which it is a bisection; the other bisection is A239872. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 EXAMPLE a(7) counts these 9 partitions of 15:  [12,1,2], [10,1,4], [10,3,2], [4,9,2], [8,1,6], [8,5,2], [8,3,4], [6,7,2], [6,5,4]. MAPLE b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or       abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+       `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))     end: a:= n-> b(2*n+1\$2, 1): seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014 MATHEMATICA d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p[n_] := p[n] = Select[d[n], Count[#, _?OddQ] == -1 + Count[#, _?EvenQ] &]; t = Table[p[n], {n, 0, 20}] TableForm[t] (* shows the partitions *) u = Table[Length[p[2 n + 1]], {n, 0, 38}]  (* A239873 *) (* Peter J. C. Moses, Mar 10 2014 *) b[n_, i_, t_] := b[n, i, t] = If[n > i (i + 1)/2 || Abs[t] > n, 0, If[n == 0, 1, b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + (2 Mod[i, 2] - 1)]]]]; a[n_] := b[2n+1, 2n+1, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *) CROSSREFS Cf. A239241, A239871, A239872, A239832. Sequence in context: A224812 A194256 A194246 * A224811 A024617 A025698 Adjacent sequences:  A239870 A239871 A239872 * A239874 A239875 A239876 KEYWORD nonn,easy AUTHOR Clark Kimberling, Mar 29 2014 STATUS approved

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Last modified May 18 19:58 EDT 2021. Contains 344002 sequences. (Running on oeis4.)