1,7

Related to the 'number of sums containing k' phenomena reported at link. Define P_k(n,j) to be the number of partitions of n with minimum part j and containing k, P_k(n) as the number of partitions of n that contain k as a part and P(n,j) as the number of partitions of k that have minimum part k, then: P_k(n)=sum{i=1,k-1,P_k(n-i,i)}+P(n-k,k) which (unproved) gives P(n-k). This sequence gives P_3(n,2). E.g. assume P_3(9)=11. P_3(10)=P_3(9,1)+P_3(8,2)+P(7,3)=11+2+2=15, where P(7,3) is given by A008483(7).

Jon Perry, Partition Function

A002865(n) = a(n+2). - James Sellers, Dec 06 2005.

See A002865.

a(3): 3

a(5): 2+3

a(6): 3+3

a(7): 2+2+3, 3+4

a(8): 2+3+3, 3+5

a(9): 2+3+4, 2+2+2+3, 3+3+3, 3+6

a(10): 2+3+5, 2+2+3+3, 3+7, 3+3+4

a(11): 2+2+3+4, 2+3+6, 2+2+2+2+3, 2+3+3+3, 3+4+4, 3+8, 3+3+5,

a(12): 2+2+2+3+3, 2+3+3+4, 2+3+7, 2+2+3+5, 3+9, 3+3+6, 3+4+5, 3+3+3+3

a(13): 2+2+2+2+2+3, 2+2+2+3+4, 2+2+3+6, 2+2+3+3+3, 2+3+4+4, 2+3+3+5,

2+3+8, 3+10, 3+3+7, 3+4+6, 3+5+5, 3+3+3+4

Needs["DiscreteMath`Combinatorica`"]; f[n_] := Block[{c = 0, k = 1, m = PartitionsP[n], p = Partitions[n]}, While[k < m, If[ Count[ p[[k]], 3] > 0 && Count[ p[[k]], 1] == 0, c++ ]; k++ ]; c]; Table[ f[n], {n, 1, 53}]

Cf. A008483, A002865. Essentially the same as A002865.

Sequence in context: A035989 A036000 A002865 this_sequence A014810 A026929 A035554

Adjacent sequences: A085808 A085809 A085810 this_sequence A085812 A085813 A085814

nonn

Jon Perry (perry(AT)globalnet.co.uk), Jul 25 2003

Edited, corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com),