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A240863

Number of partitions of n into distinct parts of which the number of odd parts is a part.
(history; published version)

#9 by Andrey Zabolotskiy at Fri Jul 26 06:40:40 EDT 2024

STATUS

proposed

approved

#8 by Alois P. Heinz at Fri Jul 26 04:57:37 EDT 2024

STATUS

editing

proposed

#7 by Alois P. Heinz at Fri Jul 26 04:57:34 EDT 2024

NAME

Number of partitions of n into distinct parts of which the number of odd parts is a part.

STATUS

proposed

editing

#6 by Jason Yuen at Fri Jul 26 04:55:52 EDT 2024

STATUS

editing

proposed

#5 by Jason Yuen at Fri Jul 26 04:55:47 EDT 2024

CROSSREFS

Cf. A240862, A240864, A240865, A240866, A240867, A204868A240868; for analogous sequences for unrestricted partitions, see A240573-A240579.

STATUS

approved

editing

#4 by N. J. A. Sloane at Tue Apr 22 01:30:19 EDT 2014

STATUS

proposed

approved

#3 by Clark Kimberling at Mon Apr 21 16:30:53 EDT 2014

STATUS

editing

proposed

#2 by Clark Kimberling at Mon Apr 14 15:55:12 EDT 2014

NAME

allocated for Clark Kimberling Number of partitions of n into distinct parts of which the number of odd parts is a part.

DATA

0, 1, 0, 1, 0, 1, 1, 2, 1, 3, 3, 5, 4, 7, 7, 11, 10, 15, 15, 22, 22, 31, 31, 42, 43, 58, 59, 78, 82, 105, 109, 139, 146, 183, 193, 239, 255, 311, 331, 402, 430, 516, 553, 659, 710, 839, 904, 1061, 1146, 1337, 1446, 1679, 1819, 2099, 2276, 2615, 2838, 3246

OFFSET

0,8

FORMULA

a(n) + A240870(n) = A000009(n) for n >= 0.

EXAMPLE

a(10) counts these 3 partitions: 721, 532, 4321.

MATHEMATICA

z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];

t1 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]]], {n, 0, z}] (* A240862 *)

t2 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240863, *)

t3 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]] && MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240864 *)

t4 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]] || MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240865 *)

t5 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]] && ! MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240866 *)

t6 = Table[Count[f[n], p_ /; ! MemberQ[p, Count[Mod[p, 2], 0]] && MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240867 *)

t7 = Table[Count[f[n], p_ /; ! MemberQ[p, Count[Mod[p, 2], 0]] && ! MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240868 *)

CROSSREFS

Cf. A240862, A240864, A240865, A240866, A240867, A204868; for analogous sequences for unrestricted partitions, see A240573-A240579.

KEYWORD

allocated

nonn,easy

AUTHOR

Clark Kimberling, Apr 14 2014

STATUS

approved

editing

Discussion

Mon Apr 21

16:19

OEIS Server: This sequence has not been edited or commented on for a week
yet is not proposed for review.  If it is ready for review, please
visit https://oeis.org/draft/A240863 and click the button that reads
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Thanks.
  - The OEIS Server

#1 by Clark Kimberling at Sun Apr 13 16:59:23 EDT 2014

NAME

allocated for Clark Kimberling

KEYWORD

allocated

STATUS

approved