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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 Alois P. Heinz, Table of n, a(n) for n = 0..1000 H. Bottomley, Illustration for A000009, A000041, A047967. 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 Cf. A038348, A261982. Column k=1 of A320264. Cf. A324847, A324929, A324966, A324967. Sequence in context: A004397 A324368 A241654 * A282893 A256912 A134591 Adjacent sequences: A047964 A047965 A047966 * A047968 A047969 A047970 KEYWORD nonn AUTHOR N. J. A. Sloane STATUS approved

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