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

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

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

Number of conjugacy classes of n X n matrices over GF(2). Cf. Morrison link, section 2.9. - Geoffrey Critzer, May 26 2021

Links

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, Journal of Algebra, Vol. 309, No. 2 (2007), 654-671, arXiv:math/0609262.

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, and 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) = 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

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

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

Example

From Joerg Arndt, Apr 28 2013: (Start)
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  ]
(End)

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<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

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: A296626 A364466 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