A118267 Number of partitions of n such that if the smallest part is k, then both k and k+1 occur exactly once.

0, 0, 1, 0, 1, 1, 2, 1, 4, 3, 5, 6, 9, 9, 15, 15, 22, 26, 34, 38, 53, 60, 77, 91, 115, 133, 170, 196, 243, 287, 349, 408, 500, 582, 701, 822, 984, 1147, 1371, 1594, 1889, 2204, 2596, 3014, 3549, 4111, 4812, 5576, 6502, 7512, 8744, 10081, 11691, 13470, 15573, 17898

Offset

1,7

Comments

Also number of partitions of n such that if the largest part is k, then k-1 occurs exactly once and k-2 also occurs (0 is considered to be a part of each partition). Example: a(11)=5 because we have [4,3,2,2], [4,3,2,1,1], [3,3,2,1,1,1], [2,2,2,2,2,1] and [3,2,1,1,1,1,1,1].

Links

Table of n, a(n) for n=1..56.

Formula

G.f.: sum(x^(2k+1)/product(1-x^j, j=k+2..infinity), k=1..infinity). G.f.: sum(x^(3k-3)/[(1-x^k)*product(1-x^j, j=1..k-2)], k=2..infinity).

a(n) = -p(n+4)+2*p(n+3)-p(n+1)-p(n-1)+p(n-2), where p(n) = A000041(n). - Mircea Merca, Jul 10 2013

Example

a(11)=5 because we have [8,2,1], [6,5], [6,3,2], [5,3,2,1] and [4,4,2,1].

Maple

g:=sum(x^(3*k-3)/(1-x^k)/product(1-x^j, j=1..k-2), k=2..30): gser:=series(g, x, 65): seq(coeff(gser, x, n), n=1..62);

Crossrefs

Sequence in context: A328676 A269595 A055176 * A324755 A259019 A262663

Adjacent sequences: A118264 A118265 A118266 * A118268 A118269 A118270

Keyword

nonn

Author

Emeric Deutsch, Apr 20 2006

Status

approved