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%I M0645 N0239 #82 Aug 27 2022 18:33:17
%S 0,0,1,1,2,3,5,7,10,14,20,27,37,49,66,86,113,146,190,242,310,392,497,
%T 623,782,973,1212,1498,1851,2274,2793,3411,4163,5059,6142,7427,8972,
%U 10801,12989,15572,18646,22267,26561,31602,37556,44533,52743,62338,73593
%N One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes.
%C Also number of cycle types of odd permutations.
%C Also number of partitions of n with an odd number of even parts. There is no restriction on the odd parts. - _N. Sato_, Jul 20 2005. E.g., a(6)=5 because we have [6],[4,1,1],[3,2,1],[2,2,2] and [2,1,1,1,1]. - _Emeric Deutsch_, Mar 02 2006
%C Also number of partitions of n with largest part not congruent to n modulo 2: a(2*n)=A027193(2*n), a(2*n+1)=A027187(2*n+1); a(n)=A000041(n)-A046682(n). - _Reinhard Zumkeller_, Apr 22 2006
%C From _Gus Wiseman_, Mar 31 2022: (Start)
%C Also the number of integer partitions of n with Heinz number greater than that of their conjugate, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are ranked by A352490. The complement is counted by A046682. For example, the a(n) partitions for n = 2...8 are:
%C (11) (111) (211) (221) (222) (331) (2222)
%C (1111) (2111) (2211) (2221) (3221)
%C (11111) (3111) (3211) (3311)
%C (21111) (22111) (22211)
%C (111111) (31111) (32111)
%C (211111) (41111)
%C (1111111) (221111)
%C (311111)
%C (2111111)
%C (11111111)
%C Also the number of integer partitions of n with Heinz number less than that of their conjugate, ranked by A352487. For example, the a(n) partitions for n = 2...8 are:
%C (2) (3) (4) (5) (6) (7) (8)
%C (31) (32) (33) (43) (44)
%C (41) (42) (52) (53)
%C (51) (61) (62)
%C (411) (322) (71)
%C (421) (422)
%C (511) (431)
%C (521)
%C (611)
%C (5111)
%C (End)
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A000701/b000701.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H Brian Hopkins and Michael A. Jones, <a href="https://doi.org/10.37236/1106">Shift-Induced Dynamical Systems on Partitions and Compositions</a>, Electron. J. Combin. 13 (2006), Research Paper 80, see p. 10.
%H M. Osima, <a href="http://dx.doi.org/10.4153/CJM-1952-034-x">On the irreducible representations of the symmetric group</a>, Canad. J. Math., 4 (1952), 381-384.
%F a(n) = (A000041(n) - A000700(n))/2.
%F From _Bill Gosper_, Aug 08 2005: (Start)
%F Sum a(n) q^n = q^2 + q^3 + 2 q^4 + 3 q^5 + 5 q^6 + 7 q^7 + ...
%F = -( Sum_{n>=1} (-q^2)^(n^2) ) / ( Sum_{ n = -oo..oo } (-1)^n q^(n(3n-1)/2) )
%F = (- q; q)_{oo} Sum_{n>=1} q^(2(2n-1))/(q^2;q^2)_{2n-1}
%F = (1/(q;q)_oo - 1/(q;-q)_oo)/2
%F = (1/(q;q)_oo - (-q;q^2)_oo)/2
%F = Sum{k>=0} ( 1/((q;q)_k)^2 - 1/(q^2;q^2)_k ) q^(k^2)/2
%F using the "q-Pochhammer" notation (a;q)_n := Product_{k=0..n-1} (1 - a*q^k).
%F (End)
%F a(n) = p(n-2) - p(n-8) + p(n-18) - p(n-32) + ... + (-1)^(k+1)*p(n-2*k^2) + ..., where p() is A000041(). E.g., a(20) = p(18) - p(12) + p(2) = 385 - 77 + 2 = 310. - _Vladeta Jovovic_, Aug 08 2004
%F G.f.: (1/2)*(1 - Product_{j>=1} (1-x^(2j))/(1+x^(2j)))/Product_{j>=1} (1 - x^j). - _Emeric Deutsch_, Mar 02 2006
%F a(2*n) = A236559(n). a(2*n + 1) = A236914(n). - _Michael Somos_, Aug 25 2015
%F a(n) = A330644(n)/2. - _Omar E. Pol_, Jan 10 2020
%F a(n) = A000041(n) - A046682(n) = A046682(n) - A000700(n). - _Gus Wiseman_, Mar 31 2022
%e G.f. = x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 14*x^9 + ...
%p with(combinat); A000701 := n->(numbpart(n)-A000700(n))/2;
%t a41 = PartitionsP; a700[n_] := SeriesCoefficient[ Product[1 + x^k, {k, 1, n, 2}], {x, 0, n}]; a[0] = 0; a[n_] := (a41[n] - a700[n])/2; Table[a[n], {n, 0, 48}] (* _Jean-François Alcover_, Feb 21 2012, after first formula *)
%t a[ n_] := SeriesCoefficient[ (1 / QPochhammer[ x] - 1 / QPochhammer[ x, -x]) / 2, {x, 0, n}]; (* _Michael Somos_, Aug 25 2015 *)
%t a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, x^2]) / (2 QPochhammer[ x]), {x, 0, n}]; (* _Michael Somos_, Aug 25 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] Sum[ x^(2 k) / QPochhammer[ x^2, x^2, k], {k, 1, n/2, 2}], {x, 0, n}] (* _Michael Somos_, Aug 25 2015 *)
%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (1 / QPochhammer[ x, x, k]^2 - 1 / QPochhammer[ x^2, x^2, k]) x^k^2, {k, Sqrt@n}] / 2, {x, 0, n}]]; (* _Michael Somos_, Aug 25 2015 *)
%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t Table[Length[Select[IntegerPartitions[n],Times@@Prime/@#>Times@@Prime/@conj[#]&]],{n,0,15}] (* _Gus Wiseman_, Mar 31 2022 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x^2 + A)^2 / eta(x^4 + A) ) / (2 * eta(x + A)), n))}; /* _Michael Somos_, Aug 25 2015 */
%o (PARI) q='q+O('q^60); concat([0, 0], Vec((1-eta(q^2)^2/eta(q^4))/(2*eta(q)))) \\ _Altug Alkan_, Sep 26 2018
%Y Cf. A000041, A027187, A027193, A046682, A118302, A236559, A236914.
%Y A000700 counts self-conjugate partitions, ranked by A088902.
%Y A330644 counts non-self-conjugate partitions, ranked by A352486.
%Y Heinz number (rank) and partition:
%Y - A122111 = rank of conjugate.
%Y - A296150 = parts of partition, conjugate A321649.
%Y - A352487 = rank less than conjugate, counted by A000701.
%Y - A352488 = rank greater than or equal to conjugate, counted by A046682.
%Y - A352489 = rank less than or equal to conjugate, counted by A046682.
%Y - A352490 = rank greater than conjugate, counted by A000701.
%Y - A352491 = rank minus conjugate.
%Y Cf. A114088, A115994, A171966, A238352, A258116, A325039.
%K nonn,easy,nice
%O 0,5
%A _N. J. A. Sloane_
%E Better description and more terms from _Christian G. Bower_, Apr 27 2000
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