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 A240447 Number of partitions of 2n such that (sum of parts having multiplicity 1) = sum of all other parts. 5
 1, 0, 1, 1, 3, 2, 8, 5, 18, 19, 39, 34, 105, 74, 183, 202, 381, 344, 818, 684, 1459, 1499, 2662, 2578, 5279, 4756, 8835, 9287, 15655, 15538, 28319, 27178, 46709, 49166, 78303, 80747, 135134, 134945, 216255, 231483, 353557, 369918, 581337, 600500, 915010, 987925 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The number of partitions of 2n+1 such that (sum of parts having multiplicity 1) = sum of all other parts is 0; see the Mathematica program at A240448 for related sequences. LINKS Manfred Scheucher and Alois P. Heinz, Table of n, a(n) for n = 0..650 (first 64 terms from Manfred Scheucher) Manfred Scheucher, C Code EXAMPLE a(6) counts these 8 partitions of 12:  633, 6222, 62211, 6111111, 5331, 52221, 4332, 42111111 . MAPLE f := proc(L, p)     a := 0 ;     for i in L do         if i = p then             a := a+1 ;         end if;     end do:     a; end proc: sp1 := proc(L)     a1 :=0 ;     ao :=0 ;     for i in L do         if f(L, i) = 1 then             a1 := a1+i;         else             ao := ao+i;         end if;     end do:     if ( a1 = ao) then         true;     else         false;     end if; end proc: A240447 := proc(n)     a := 0 ;     for p in combinat[partition](2*n) do         if sp1(p) then             a := a+1 ;         end if;     end do:     a ; end proc: # R. J. Mathar, Mar 06 2015 # second Maple program: b:= proc(n, i, s) option remember; `if`(n=0, `if`(s=0, 1, 0),       `if`(i<1 or s>0 and i*(i+1)/2n, 0,        add(b(n-i*j, i-1, s+`if`(j=1, -i, i*j)), j=0..n/i)))     end: a:= n-> b(2*n\$2, 0): seq(a(n), n=0..70);  # Alois P. Heinz, May 31 2015 MATHEMATICA ColumnForm[t = Table[Select[IntegerPartitions[n], 2 Total[First[Transpose[Select[#, Last[#] == 1 &] /. {} -> {{0, 0}}]]] &[Tally[#]] == n &], {n, 0, 30, 2}]] (* shows partitions *) Map[Length, t] (* A240447 *)  (* Peter J. C. Moses, Apr 02 2014 *) b[n_, i_, s_] := b[n, i, s] = If[n == 0, If[s == 0, 1, 0], If[i<1 || s>0 && i*(i+1)/2 < s || Abs[s] > n, 0, Sum[b[n-i*j, i-1, s + If[j == 1, -i, i*j]], {j, 0, n/i}]]]; a[n_] := b[2*n, 2*n, 0]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *) CROSSREFS Cf. A240448, A240449, A240451, A240452. Sequence in context: A126320 A322845 A307705 * A135992 A182638 A172084 Adjacent sequences:  A240444 A240445 A240446 * A240448 A240449 A240450 KEYWORD nonn,easy AUTHOR Clark Kimberling, Apr 05 2014 EXTENSIONS More terms from Manfred Scheucher, May 30 2015 STATUS approved

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Last modified January 20 20:35 EST 2020. Contains 331096 sequences. (Running on oeis4.)