This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A047967 #53 Aug 29 2023 09:44:47
%S A047967 0,0,1,1,3,4,7,10,16,22,32,44,62,83,113,149,199,259,339,436,563,716,
%T A047967 913,1151,1453,1816,2271,2818,3496,4309,5308,6502,7959,9695,11798,
%U A047967 14298,17309,20877,25151,30203,36225,43323,51748,61651,73359,87086,103254,122164
%N A047967 Number of partitions of n with some part repeated.
%C A047967 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
%C A047967 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
%C A047967 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
%H A047967 Alois P. Heinz, <a href="/A047967/b047967.txt">Table of n, a(n) for n = 0..1000</a>
%H A047967 H. Bottomley, <a href="/A000009/a000009.gif">Illustration for A000009, A000041, A047967</a>.
%F A047967 a(n) = A000041(n) - A000009(n).
%F A047967 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
%F A047967 G.f.: 1/P(x) - P(x^2)/P(x) where P(x) = Product_{k>=1} (1-x^k). - _Joerg Arndt_, Jun 21 2011
%F A047967 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
%e A047967 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).
%p A047967 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
%t A047967 Table[PartitionsP[n]-PartitionsQ[n],{n,0,50}] (* _Harvey P. Dale_, Jan 17 2019 *)
%o A047967 (PARI) x='x+O('x^66); concat([0,0], Vec(1/eta(x)-eta(x^2)/eta(x))) \\ _Joerg Arndt_, Jun 21 2011
%Y A047967 Cf. A038348, A261982.
%Y A047967 Column k=1 of A320264.
%Y A047967 Cf. A324847, A324929, A324966, A324967.
%K A047967 nonn
%O A047967 0,5
%A A047967 _N. J. A. Sloane_