A070877 -id:A070877 - cp's OEIS frontend

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

Sharon Sela (sharonsela(AT)hotmail.com), May 21 2002

nonn

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

			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)

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.

Cf. A006951, A000041, A070877.
Cf. A083355.
Column k=2 of A246935.
Cf. A048651.
Row sums of A256193.
Antidiagonal sums of A322210.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-2*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

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

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 *)

S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

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

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

Edited and extended by Robert G. Wilson v, May 25 2002

A292128 Expansion of Product_{k>=1} (1 - 3*x^k).

Original entry on oeis.org

1, -3, -3, 6, 6, 15, -12, -3, -30, -48, 6, -12, 15, 78, 186, -21, 168, 42, 42, -246, -408, -156, -399, -552, -498, -246, 213, 1248, -318, 1608, 1392, 2508, 1482, 2976, -480, -1011, 1500, -1704, -4296, -4206, -8499, -8652, -7626, -7050, -192, -13008, -480, -2118

Offset: 0

Seiichi Manyama, Sep 09 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Column k=3 of A292131.
Product_{k>=1} (1 - m*x^k): A010815 (m=1), A070877 (m=2), this sequence (m=3).
Cf. A032308, A242587, A292130.

N=66; x='x+O('x^N); Vec(prod(n=1, N, 1-3*x^n))

G.f.: Sum_{i>=0} (-3)^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 13 2018

A292131 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - k*x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -2, 0, 0, 1, -4, -3, 2, 0, 0, 1, -5, -4, 6, 2, 1, 0, 1, -6, -5, 12, 6, 6, 0, 0, 1, -7, -6, 20, 12, 15, -2, 1, 0, 1, -8, -7, 30, 20, 28, -12, 2, 0, 0, 1, -9, -8, 42, 30, 45, -36, -3, -6, 0, 0, 1, -10, -9, 56, 42, 66, -80

Offset: 0

Seiichi Manyama, Sep 09 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0, -1, -2, -3, -4, ...
   0,  0,  2,  6, 12, ...
   0,  0,  2,  6, 12, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000007, A010815, A070877, A292128.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292132.
Cf. A246935, A286957.

A292129 Triangle read by rows: T(n,k) = (-2) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, -2, 0, -2, 0, -2, 4, 0, -2, 4, 0, -2, 8, 0, -2, 8, -8, 0, -2, 12, -8, 0, -2, 12, -16, 0, -2, 16, -24, 0, -2, 16, -32, 16, 0, -2, 20, -40, 16, 0, -2, 20, -56, 32, 0, -2, 24, -64, 48, 0, -2, 24, -80, 80, 0, -2, 28, -96, 96, -32, 0, -2, 28, -112, 144, -32, 0

Offset: 0

Seiichi Manyama, Sep 09 2017

			First few rows are:
  1;
  0, -2;
  0, -2;
  0, -2,  4;
  0, -2,  4;
  0, -2,  8;
  0, -2,  8,  -8;
  0, -2, 12,  -8;
  0, -2, 12, -16;
  0, -2, 16, -24;
  0, -2, 16, -32, 16.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A070877.
Columns 0-1 give A000007, (-1)*A007395.
Cf. A003056, A291970.

A349925 Dirichlet g.f.: Product_{k>=2} (1 - 2 * k^(-s)).

Original entry on oeis.org

1, -2, -2, -2, -2, 2, -2, 2, -2, 2, -2, 6, -2, 2, 2, 2, -2, 6, -2, 6, 2, 2, -2, 2, -2, 2, 2, 6, -2, 2, -2, 6, 2, 2, 2, 2, -2, 2, 2, 2, -2, 2, -2, 6, 6, 2, -2, -2, -2, 6, 2, 6, -2, 2, 2, 2, 2, 2, -2, -6, -2, 2, 6, -2, 2, 2, -2, 6, 2, 2, -2, -6, -2, 2, 6

Offset: 1

Ilya Gutkovskiy, Dec 05 2021

sign

Cf. A070877, A114592, A226177, A339701, A349926.

A352402 Expansion of Product_{k>=1} 1 / (1 + 2^(k-1)*x^k).

Original entry on oeis.org

1, -1, -1, -3, -1, -7, -1, -15, 31, -63, 159, -95, 671, -287, 3231, -2975, 15519, -7839, 44191, -34975, 224415, -291999, 863391, -990367, 2927775, -4902047, 12561567, -27225247, 56470687, -102640799, 152153247, -422620319, 877243551, -2278272159, 3357125791

Offset: 0

Ilya Gutkovskiy, Jun 08 2022

sign

Cf. A032302, A070877, A070933, A071109, A075900, A304961, A327550, A337299, A352762.

nmax = 34; CoefficientList[Series[Product[1/(1 + 2^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[IntegerPartitions[n, {k}]] 2^(n - k), {k, 0, n}], {n, 0, 34}]

a(n) = Sum_{k=0..n} (-1)^k * p(n,k) * 2^(n-k), where p(n,k) is the number of partitions of n into k parts.

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

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A292128 Expansion of Product_{k>=1} (1 - 3*x^k).

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A292131 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - k*x^j).

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A292129 Triangle read by rows: T(n,k) = (-2) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A349925 Dirichlet g.f.: Product_{k>=2} (1 - 2 * k^(-s)).

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A352402 Expansion of Product_{k>=1} 1 / (1 + 2^(k-1)*x^k).

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