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A047967 Number of partitions of n with some part repeated. 61
0, 0, 1, 1, 3, 4, 7, 10, 16, 22, 32, 44, 62, 83, 113, 149, 199, 259, 339, 436, 563, 716, 913, 1151, 1453, 1816, 2271, 2818, 3496, 4309, 5308, 6502, 7959, 9695, 11798, 14298, 17309, 20877, 25151, 30203, 36225, 43323, 51748, 61651, 73359, 87086, 103254, 122164 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Also number of partitions of n with at least one even part. - Vladeta Jovovic, Sep 10 2003. Example: a(5)=4 because we have [4,1], [3,2], [2,2,1] and [2,1,1,1] ([5], [3,1,1] and [1,1,1,1,1] do not qualify). - Emeric Deutsch, Mar 30 2006
Also number of partitions of n (where it is assumed that the least part is 0) such that at least one difference is at least two. Example: a(5)=4 because we have [5,0], [4,1,0], [3,2,0] and [3,1,1,0] ([2,2,1,0], [2,1,1,1,0] and [1,1,1,1,1,0] do not qualify). - Emeric Deutsch, Mar 30 2006
The Heinz numbers of these partitions (with some part repeated) are given by A013929. Equivalent to Vladeta Jovovic's comment, a(n) is also the number of integer partitions whose product of parts is even. The Heinz numbers of these latter partitions are given by A324929. - Gus Wiseman, Mar 23 2019
LINKS
FORMULA
a(n) = A000041(n) - A000009(n).
G.f.: Sum_{k>=1} x^(2*k)*(Product_{j>=k+1} (1+x^j)) / Product_{j=1..k} (1-x^j) = Sum_{k>=1} x^(2*k)/(Product_{j=1..2*k} (1-x^j)*Product_{j>=k} (1-x^(2*j+1))). - Emeric Deutsch, Mar 30 2006
G.f.: 1/P(x) - P(x^2)/P(x) where P(x) = Product_{k>=1} (1-x^k). - Joerg Arndt, Jun 21 2011
a(n) = p(n-2)+p(n-4)-p(n-10)-p(n-14)+...+(-1^(j-1))*p(n-j*(3*j-1)) + (-1^(j-1))*p(n-j*(3*j+1))+..., where p(n) = A000041(n). - Gregory L. Simay, Aug 28 2023
EXAMPLE
a(5) = 4 because we have [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1] ([5], [4,1] and [3,2] do not qualify).
MAPLE
g:=sum(x^(2*k)*product(1+x^j, j=k+1..70)/product(1-x^j, j=1..k), k=1..40): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=0..44); # Emeric Deutsch, Mar 30 2006
MATHEMATICA
Table[PartitionsP[n]-PartitionsQ[n], {n, 0, 50}] (* Harvey P. Dale, Jan 17 2019 *)
PROG
(PARI) x='x+O('x^66); concat([0, 0], Vec(1/eta(x)-eta(x^2)/eta(x))) \\ Joerg Arndt, Jun 21 2011
CROSSREFS
Column k=1 of A320264.
Sequence in context: A004397 A324368 A241654 * A282893 A256912 A134591
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified April 25 03:15 EDT 2024. Contains 371964 sequences. (Running on oeis4.)