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A070933
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Expansion of Product_{k>=1} 1/(1 - 2*t^k).
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27
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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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, Poincaré 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
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FORMULA
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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<m. - Vladimir Kruchinin Sep 07 2014
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
G.f.: Sum_{i>=0} 2^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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<m then 0 else if n=m then 2 else sum(2*S(n-k, k), k, m, n/2)+2;
makelist(S(n, 1), n, 0, 17); /* Vladimir Kruchinin, Sep 07 2014 */
(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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Sharon Sela (sharonsela(AT)hotmail.com), May 21 2002
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EXTENSIONS
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Edited and extended by Robert G. Wilson v, May 25 2002
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STATUS
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approved
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