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A070933 Expansion of Product_{k>=1} 1/(1 - 2*t^k).

%I A070933 #116 Jun 29 2025 11:01:15
%S A070933 1,2,6,14,34,74,166,350,746,1546,3206,6550,13386,27114,54894,110630,
%T A070933 222794,447538,898574,1801590,3610930,7231858,14480654,28983246,
%U A070933 58003250,116054034,232186518,464475166,929116402,1858449178,3717247638,7434950062,14870628026,29742206138,59485920374,118973809798,237950730522,475905520474
%N A070933 Expansion of Product_{k>=1} 1/(1 - 2*t^k).
%C A070933 See A083355 for a similar formula. - _Thomas Wieder_, May 07 2008
%C A070933 Partitions of n into 2 sorts of parts: the parts are unordered, but not the sorts; see example and formula by Wieder. - _Joerg Arndt_, Apr 28 2013
%C A070933 Convolution inverse of A070877. - _George Beck_, Dec 02 2018
%C A070933 Number of conjugacy classes of n X n matrices over GF(2).  Cf. Morrison link, section 2.9. - _Geoffrey Critzer_, May 26 2021
%H A070933 Alois P. Heinz, <a href="/A070933/b070933.txt">Table of n, a(n) for n = 0..1000</a> (first 501 terms from T. D. Noe)
%H A070933 Dragomir Z. Djokovic, <a href="https://doi.org/10.1016/j.jalgebra.2006.09.018">Poincaré series of some pure and mixed trace algebras of two generic matrices</a>, Journal of Algebra, Vol. 309, No. 2 (2007), 654-671, arXiv:<a href="https://arxiv.org/abs/math/0609262">math/0609262</a>.
%H A070933 Kent E. Morrison, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%H A070933 Igor Pak, Greta Panova, and Damir Yeliussizov, <a href="https://arxiv.org/abs/1804.04693">On the largest Kronecker and Littlewood-Richardson coefficients</a>, arXiv:1804.04693 [math.CO], 2018.
%H A070933 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%F A070933 a(n) = (1/n)*Sum_{k=1..n} A054598(k)*a(n-k). - _Vladeta Jovovic_, Nov 23 2002
%F A070933 a(n) is asymptotic to c*2^n where c=3.46253527447396564949732... - _Benoit Cloitre_, Oct 26 2003. Right value of this constant is c = 1/A048651 = 3.46274661945506361153795734292443116454075790290443839132935303175891543974042... . - _Vaclav Kotesovec_, Sep 09 2014
%F A070933 Euler transform of A000031(n). - _Vladeta Jovovic_, Jun 23 2004
%F A070933 a(n) = Sum_{k=1..n} p(n,k)*A000079(k) where p(n,k) = number of integer partitions of n into k parts. - _Thomas Wieder_, May 07 2008
%F A070933 a(n) = S(n,1), where S(n,m) = 2 + Sum_{k=m..floor(n/2)} 2*S(n-k,k), S(n,n)=2, S(0,m)=1, S(n,m)=0 for n < m. - _Vladimir Kruchinin_, Sep 07 2014
%F A070933 a(n) = Sum_{lambda,mu,nu} (c^{lambda}_{mu,nu})^2, where lambda ranges over all partitions of n, mu and nu range over all partitions satisfying |mu| + |nu| = n, and c^{lambda}_{mu,nu} denotes a Littlewood-Richardson coefficient. - _Richard Stanley_, Nov 16 2014
%F A070933 G.f.: Sum_{i>=0} 2^i*x^i/Product_{j=1..i} (1 - x^j). - _Ilya Gutkovskiy_, Apr 12 2018
%F A070933 G.f.: Product_{j>=1} Product_{i>=1} 1/(1-x^(i*j))^A001037(j) given in Morrison link section 2.9. - _Geoffrey Critzer_, May 26 2021
%e A070933 From _Joerg Arndt_, Apr 28 2013: (Start)
%e A070933 There are a(3)=14 partitions of 3 with 2 ordered sorts. Here p:s stands for "part p of sort s":
%e A070933 01:  [ 1:0  1:0  1:0  ]
%e A070933 02:  [ 1:0  1:0  1:1  ]
%e A070933 03:  [ 1:0  1:1  1:0  ]
%e A070933 04:  [ 1:0  1:1  1:1  ]
%e A070933 05:  [ 1:1  1:0  1:0  ]
%e A070933 06:  [ 1:1  1:0  1:1  ]
%e A070933 07:  [ 1:1  1:1  1:0  ]
%e A070933 08:  [ 1:1  1:1  1:1  ]
%e A070933 09:  [ 2:0  1:0  ]
%e A070933 10:  [ 2:0  1:1  ]
%e A070933 11:  [ 2:1  1:0  ]
%e A070933 12:  [ 2:1  1:1  ]
%e A070933 13:  [ 3:0  ]
%e A070933 14:  [ 3:1  ]
%e A070933 (End)
%p A070933 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A070933       b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))
%p A070933     end:
%p A070933 a:= n-> b(n$2):
%p A070933 seq(a(n), n=0..50);  # _Alois P. Heinz_, Sep 07 2014
%t A070933 CoefficientList[ Series[ Product[1 / (1 - 2t^k), {k, 1, 35}], {t, 0, 35}], t]
%t A070933 CoefficientList[Series[E^Sum[2^k*x^k / (k*(1-x^k)), {k,1,30}],{x,0,30}],x] (* _Vaclav Kotesovec_, Sep 09 2014 *)
%t A070933 (O[x]^20 - 1/QPochhammer[2,x])[[3]] (* _Vladimir Reshetnikov_, Nov 20 2015 *)
%o A070933 (PARI) N=66; q='q+O('q^N); Vec(1/sum(n=0, N, (-2)^n*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) )) \\ _Joerg Arndt_, Mar 09 2014
%o A070933 (Maxima)
%o A070933 S(n,m):=if n=0 then 1 else if n<m then 0 else if n=m then 2 else sum(2*S(n-k,k),k,m,n/2)+2;
%o A070933 makelist(S(n,1),n,0,17); /* _Vladimir Kruchinin_, Sep 07 2014 */
%o A070933 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-2*x^k): k in [1..m]]) )); // _G. C. Greubel_, Oct 31 2018
%Y A070933 Cf. A006951, A000041, A070877.
%Y A070933 Cf. A083355.
%Y A070933 Column k=2 of A246935.
%Y A070933 Cf. A048651.
%Y A070933 Row sums of A256193.
%Y A070933 Antidiagonal sums of A322210.
%K A070933 nonn
%O A070933 0,2
%A A070933 Sharon Sela (sharonsela(AT)hotmail.com), May 21 2002
%E A070933 Edited and extended by _Robert G. Wilson v_, May 25 2002