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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, 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 20 02:48 EST 2022. Contains 350467 sequences. (Running on oeis4.)