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 A000701 One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes. (Formerly M0645 N0239) 25
 0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 146, 190, 242, 310, 392, 497, 623, 782, 973, 1212, 1498, 1851, 2274, 2793, 3411, 4163, 5059, 6142, 7427, 8972, 10801, 12989, 15572, 18646, 22267, 26561, 31602, 37556, 44533, 52743, 62338, 73593 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also number of cycle types of odd permutations. 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 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 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe) Brian Hopkins, Michael A. Jones, Shift-Induced Dynamical Systems on Partitions and Compositions, Electron. J. Combin. 13 (2006), Research Paper 80, see p. 10. M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384. FORMULA a(n) = (A000041(n) - A000700(n))/2. From Bill Gosper, Aug 08 2005: (Start) Sum a(n) q^n = q^2 + q^3 + 2 q^4 + 3 q^5 + 5 q^6 + 7 q^7 + ... = -( Sum_{n = 1 .. oo} (-q^2)^(n^2) ) / ( Sum_{ n = -oo, oo } (-1)^n q^(n(3n-1)/2) ) = (- q; q)_{oo} Sum_{n=1..oo} q^(2(2n-1))/(q^2;q^2)_{2n-1} = (1/(q;q)_oo - 1/(q;-q)_oo)/2 = (1/(q;q)_oo - (-q;q^2)_oo)/2 = Sum{ k = 0..oo } ( 1/((q;q)_k)^2 - 1/(q^2;q^2)_k ) q^(k^2)/2 using the "q-pochhammer" notation (a;q)_n := Product_{k=0..n-1} 1-a*q^k. (End) 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 G.f.: (1/2)(1-product((1-x^(2j))/(1+x^(2j)), j=1..oo))/product(1-x^j, j=1..oo). - Emeric Deutsch, Mar 02 2006 a(2*n) = A236559(n). a(2*n + 1) = A236914(n). - Michael Somos, Aug 25 2015 EXAMPLE G.f. = x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 14*x^9 + ... MAPLE with(combinat); A000701 := n->(numbpart(n)-A000700(n))/2; MATHEMATICA 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 *) a[ n_] := SeriesCoefficient[ (1 / QPochhammer[ x] - 1 / QPochhammer[ x, -x]) / 2, {x, 0, n}]; (* Michael Somos, Aug 25 2015 *) a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, x^2]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Aug 25 2015 *) 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 *) 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 *) PROG (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 */ (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 CROSSREFS Cf. A000041, A000700, A027187, A027193, A046682, A236559, A236914. Cf. A118302. Sequence in context: A036005 A104503 A027340 * A123975 A321728 A214077 Adjacent sequences:  A000698 A000699 A000700 * A000702 A000703 A000704 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Better description and more terms from Christian G. Bower, Apr 27 2000 STATUS approved

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Last modified January 16 15:31 EST 2019. Contains 319195 sequences. (Running on oeis4.)