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 A236634 Number of unbalanced partitions of n: the largest part is not equal to the number of parts. 2
 0, 2, 2, 4, 6, 10, 12, 20, 26, 38, 50, 70, 90, 124, 160, 212, 272, 356, 450, 582, 732, 932, 1166, 1470, 1824, 2280, 2814, 3486, 4280, 5268, 6428, 7864, 9552, 11614, 14044, 16990, 20450, 24626, 29524, 35392, 42272, 50472, 60060, 71444, 84734, 100432, 118736 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of partitions of n whose rank is not 0. LINKS Table of n, a(n) for n=1..47. FORMULA a(n) = A000041(n) - A047993(n) = 2*A064173(n). EXAMPLE For n = 5 we have: ------------------------------------------------------- Partitions Largest Number Dyson's of 5 part of parts rank Type ------------------------------------------------------- 5 5 - 1 = 4 unbalanced 4+1 4 - 2 = 2 unbalanced 3+2 3 - 2 = 1 unbalanced 3+1+1 3 - 3 = 0 balanced 2+2+1 2 - 3 = -1 unbalanced 2+1+1+1 2 - 4 = -2 unbalanced 1+1+1+1+1 1 - 5 = -4 unbalanced ------------------------------------------------------- There are 6 partitions whose rank is not 0, so a(5) = 6. MATHEMATICA P = PartitionsP; a[n_] := P[n] - Sum[-(-1)^k (P[n - (3k^2 - k)/2] - P[n - (3k^2 + k)/2]), {k, 1, Floor[(1 + Sqrt[1 + 24n])/6]}]; a /@ Range[46] (* Jean-François Alcover, Jan 11 2020, after Wouter Meeussen in A047993 *) CROSSREFS Cf. A000041, A047993, A064173, A209616. Sequence in context: A326528 A326633 A323587 * A034406 A098330 A240310 Adjacent sequences: A236631 A236632 A236633 * A236635 A236636 A236637 KEYWORD nonn AUTHOR Omar E. Pol, Feb 18 2014 STATUS approved

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Last modified May 26 21:41 EDT 2024. Contains 372847 sequences. (Running on oeis4.)