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 A070933 Expansion of Product_{k>=1} 1/(1 - 2*t^k). 27
 1, 2, 6, 14, 34, 74, 166, 350, 746, 1546, 3206, 6550, 13386, 27114, 54894, 110630, 222794, 447538, 898574, 1801590, 3610930, 7231858, 14480654, 28983246, 58003250, 116054034, 232186518, 464475166, 929116402, 1858449178, 3717247638, 7434950062, 14870628026, 29742206138, 59485920374, 118973809798, 237950730522, 475905520474 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS See A083355 for a similar formula. - Thomas Wieder, May 07 2008 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 Convolution inverse of A070877. - George Beck, Dec 02 2018 LINKS T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from T. D. Noe) Dragomir Z. Djokovic, PoincarĂ© series of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1. Igor Pak, Greta Panova, Damir Yeliussizov, On the largest Kronecker and Littlewood-Richardson coefficients, arXiv:1804.04693 [math.CO], 2018. N. J. A. Sloane, Transforms FORMULA a(n) = (1/n)*Sum_{k=1..n} A054598(k)*a(n-k). - Vladeta Jovovic, Nov 23 2002 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 Euler transform of A000031(n). - Vladeta Jovovic, Jun 23 2004 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 a(n) = S(n,1), where S(n,m)= sum(k=m..n/2, 2*S(n-k,k))+2, S(n,n)=2, S(0,m)=1, S(n,m)=0 for n=0} 2^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018 EXAMPLE There are a(3)=14 partitions of 3 with 2 ordered sorts. Here p:s stands for "part p of sort s": 01:  [ 1:0  1:0  1:0  ] 02:  [ 1:0  1:0  1:1  ] 03:  [ 1:0  1:1  1:0  ] 04:  [ 1:0  1:1  1:1  ] 05:  [ 1:1  1:0  1:0  ] 06:  [ 1:1  1:0  1:1  ] 07:  [ 1:1  1:1  1:0  ] 08:  [ 1:1  1:1  1:1  ] 09:  [ 2:0  1:0  ] 10:  [ 2:0  1:1  ] 11:  [ 2:1  1:0  ] 12:  [ 2:1  1:1  ] 13:  [ 3:0  ] 14:  [ 3:1  ] - Joerg Arndt, Apr 28 2013 MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,       b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))     end: a:= n-> b(n\$2): seq(a(n), n=0..50);  # Alois P. Heinz, Sep 07 2014 MATHEMATICA CoefficientList[ Series[ Product[1 / (1 - 2t^k), {k, 1, 35}], {t, 0, 35}], t] CoefficientList[Series[E^Sum[2^k*x^k / (k*(1-x^k)), {k, 1, 30}], {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 09 2014 *) (O[x]^20 - 1/QPochhammer[2, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *) PROG (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 (Maxima) S(n, m):=if n=0 then 1 else if n:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-2*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018 CROSSREFS Cf. A006951, A000041, A070877. Cf. A083355. Column k=2 of A246935. Cf. A048651. Row sums of A256193. Antidiagonal sums of A322210. Sequence in context: A124613 A296626 A124614 * A059570 A208902 A018016 Adjacent sequences:  A070930 A070931 A070932 * A070934 A070935 A070936 KEYWORD nonn AUTHOR Sharon Sela (sharonsela(AT)hotmail.com), May 21 2002 EXTENSIONS Edited and extended by Robert G. Wilson v, May 25 2002 STATUS approved

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Last modified April 13 18:45 EDT 2021. Contains 342939 sequences. (Running on oeis4.)