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%I M0524 #184 Mar 12 2021 22:24:41
%S 1,1,1,2,3,4,5,7,10,13,16,21,28,35,43,55,70,86,105,130,161,196,236,
%T 287,350,420,501,602,722,858,1016,1206,1431,1687,1981,2331,2741,3206,
%U 3740,4368,5096,5922,6868,7967,9233,10670,12306,14193,16357,18803,21581
%N G.f.: Product_{k>=1} (1 + x^(2*k - 1)) / (1 - x^(2*k)).
%C Also the number of partitions of n in which all odd parts are distinct. There is no restriction on the even parts. E.g., a(9)=13 because "9 = 8+1 = 7+2 = 6+3 = 6+2+1 = 5+4 = 5+3+1 = 5+2+2 = 4+4+1 = 4+3+2 = 4+2+2+1 = 3+2+2+2 = 2+2+2+2+1". - _Noureddine Chair_, Feb 03 2005
%C Number of partitions of n in which each even part occurs with even multiplicity. There is no restriction on the odd parts.
%C Also the number of partitions of n into parts not congruent to 2 mod 4. - _James A. Sellers_, Feb 08 2002
%C Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras o(n) of skew-symmetric n X n matrices, n=0,1,2,3,... (the cases n=0,1 being degenerate). This sequence, A015128 and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
%C Poincaré series [or Poincare series] (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) ⊗ E(e_1, e_2, ... e_n) with b_i 2-dimensional, e_i one-dimensional and the permutation action of S_n, in the case n=2.
%C Equals polcoeff inverse of A010054 with alternate signs. - _Gary W. Adamson_, Mar 15 2010
%C It appears that this sequence is related to the generalized hexagonal numbers (A000217) in the same way as the partition numbers A000041 are related to the generalized pentagonal numbers A001318. (See the table in comments section of A195825.) Conjecture: this is 1 together with the row sums of triangle A195836, also column 1 of A195836, also column 2 of the square array A195825. - _Omar E. Pol_, Oct 09 2011
%C Since this is also column 2 of A195825 so the sequence contains only one plateau [1, 1, 1] of level 1 and length 3. For more information see A210843. - _Omar E. Pol_, Jun 27 2012
%C Convolution of A035363 and A000700. - _Vaclav Kotesovec_, Aug 17 2015
%C Also the number of ways to stack n triangles in a valley (pointing upwards or downwards depending on row parity). - _Seiichi Manyama_, Jul 07 2018
%D A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 108.
%D M. D. Hirschhorn, The Power of q, Springer, 2017. See pod, page 297.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A006950/b006950.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%H N. Chair, <a href="https://arxiv.org/abs/hep-th/0409011">Partition identities from Partial Supersymmetry</a>, arXiv:hep-th/0409011, 2004.
%H Brian Drake, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.020">Limits of areas under lattice paths</a>, Discrete Math. 309 (2009), no. 12, 3936-3953.
%H Luca Ferrari, <a href="https://arxiv.org/abs/1606.06624">Schröder partitions, Schröder tableaux and weak poset patterns</a>, arXiv:1606.06624 [math.CO], 2016. Mentions this sequence.
%H Mircea Merca, <a href="https://dx.doi.org/10.1016/j.jnt.2016.12.015">New relations for the number of partitions with distinct even parts</a>, Journal of Number Theory 176 (July 2017), 1-12.
%H Victor S. Miller, <a href="https://arxiv.org/abs/1606.09299">Counting Matrices that are Squares</a>, arXiv:1606.09299 [math.GR], 2016.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Maxie D. Schmidt, <a href="https://arxiv.org/abs/1705.03488">Exact Formulas for the Generalized Sum-of-Divisors Functions</a>, arXiv:1705.03488 [math.NT], 2017. See Example 4.2 p. 13.
%H Andrew Sills, <a href="https://works.bepress.com/andrew_sills/40/">Rademacher-Type Formulas for Restricted Partition and Overpartition Functions</a>, Ramanujan Journal, 23 (1-3): 253-264, 2010.
%H L. Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Wang2/wang31.html">New Congruences for Partitions where the Odd Parts are Distinct</a>, J. Int. Seq. 18 (2015) # 15.4.2.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H M. P. Zaletel and R. S. K. Mong, <a href="https://arxiv.org/abs/1208.4862">Exact Matrix Product States for Quantum Hall Wave Functions</a>, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012
%F a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1)*A002129(k)*a(n-k), n > 1, a(0)=1. - _Vladeta Jovovic_, Feb 05 2002
%F G.f.: 1/Sum_{k>=0} (-x)^(k*(k+1)/2). - _Vladeta Jovovic_, Sep 22 2002 [corrected by _Vaclav Kotesovec_, Aug 17 2015]
%F a(n) = A059777(n-1)+A059777(n), n > 0. - _Vladeta Jovovic_, Sep 22 2002
%F G.f.: Product_{m>=1} (1+x^m)^(if A001511(m) > 1, A001511(m)-1 else A001511(m)). - _Jon Perry_, Apr 15 2005
%F Expansion of 1 / psi(-x) in powers of x where psi() is a Ramanujan theta function.
%F Expansion of q^(1/8) * eta(q^2) / (eta(q) * eta(q^4)) in powers of q.
%F Convolution inverse of A106459. - _Michael Somos_, Nov 02 2005
%F G.f.: exp( Sum_{n>=1} [Sum_{d|n} (-1)^(n-d)*d] * x^n/n ). - _Paul D. Hanna_, Jul 22 2009
%F a(n) ~ (8*n+1) * cosh(sqrt(8*n-1)*Pi/4) / (16*sqrt(2)*n^2) - sinh(sqrt(8*n-1)*Pi/4) / (2*Pi*n^(3/2)) ~ exp(Pi*sqrt(n/2))/(4*sqrt(2)*n) * (1 - (2/Pi + Pi/16)/sqrt(2*n) + (3/16 + Pi^2/1024)/n). - _Vaclav Kotesovec_, Aug 17 2015, extended Jan 09 2017
%F Can be computed recursively by Sum_{j>=0} (-1)^(ceiling(j/2)) a(n - j(j+1)/2) = 0, for n > 0. [Merca, Theorem 4.3] - _Eric M. Schmidt_, Sep 21 2017
%F a(n) = A000041(n) - A085642(n), for n >= 1. - _Wouter Meeussen_, Dec 20 2017
%e G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 13*x^9 + ...
%e G.f. = q^-1 + q^7 + q^15 + 2*q^23 + 3*q^31 + 4*q^39 + 5*q^47 + 7*q^55 + 10*q^63 + ...
%e From _Seiichi Manyama_, Jul 07 2018: (Start)
%e n | the ways to stack n triangles in a valley
%e --+------------------------------------------------------
%e 1 | *---*
%e | \ /
%e | *
%e |
%e 2 | *
%e | / \
%e | *---*
%e | \ /
%e | *
%e |
%e 3 | *---* *---*
%e | / \ / \ / \
%e | *---* *---*
%e | \ / \ /
%e | * *
%e |
%e 4 | * *
%e | / \ / \
%e | *---* *---*---* *---*
%e | / \ / \ / \ / \ / \
%e | *---* *---* *---*
%e | \ / \ / \ /
%e | * * *
%e |
%e 5 | *---* * * *---*
%e | / \ / / \ / \ \ / \
%e | *---* *---*---* *---*---* *---*
%e | / \ / \ / \ / \ / \ / \ / \
%e | *---* *---* *---* *---*
%e | \ / \ / \ / \ /
%e | * * * *
%e |
%e 6 | *
%e | / \
%e | *---* *---* * * *---*
%e | / \ / / \ / / \ / \ \ / \
%e | *---* *---*---* *---*---* *---*---*
%e | / \ / \ / \ / \ / \ / \ / \ /
%e | *---* *---* *---* *---*
%e | \ / \ / \ / \ /
%e | * * * *
%e | *
%e | / \
%e | *---*
%e | \ / \
%e | *---*
%e | \ / \
%e | *---*
%e | \ /
%e | *
%e (End)
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1)+`if`(i>n, 0, b(n-i, i-irem(i, 2)))))
%p end:
%p a:= n-> b(n, n):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jan 06 2013
%t CoefficientList[ Series[ Product[(1 + x^(2k - 1))/(1 - x^(2k)), {k, 25}], {x, 0, 50}], x] (* _Robert G. Wilson v_, Jun 28 2012 *)
%t CoefficientList[Series[x*QPochhammer[-1/x, x^2] / ((1+x)*QPochhammer[x^2, x^2]), {x, 0, 50}], x] (* _Vaclav Kotesovec_, Aug 17 2015 *)
%t CoefficientList[Series[2*(-x)^(1/8) / EllipticTheta[2, 0, Sqrt[-x]], {x, 0, 50}], x] (* _Vaclav Kotesovec_, Aug 17 2015 *)
%t b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i-Mod[i, 2]]]]];
%t a[n_] := b[n, n];
%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Dec 11 2018, after _Alois P. Heinz_ *)
%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, (-1)^(m-d)*d)*x^m/m)+x*O(x^n)), n)} \\ _Paul D. Hanna_, Jul 22 2009
%o (GWbasic)' A program with two A-numbers (Note that here A000217 are the generalized hexagonal numbers):
%o 10 Dim A000217(100), A057077(100), a(100): a(0)=1
%o 20 For n = 1 to 51: For j = 1 to n
%o 30 If A000217(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A000217(j))
%o 40 Next j: Print a(n-1);: Next n
%o # _Omar E. Pol_, Jun 10 2012
%Y See also Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%Y Cf. A015128, A046682, A106459.
%Y Cf. A163203.
%Y Cf. A010054, A085642, A316384.
%K nonn
%O 0,4
%A _N. J. A. Sloane_, _Warren D. Smith_
%E G.f. and more terms from _Vladeta Jovovic_, Feb 05 2002
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