A102866 -id:A102866 - cp's OEIS frontend

A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.

1, 1, 2, 5, 9, 19, 37, 71, 133, 252, 464, 851, 1547, 2787, 4985, 8862, 15639, 27446, 47909, 83168, 143691, 247109, 423082, 721360, 1225119, 2072762, 3494359, 5870717, 9830702, 16409939, 27309660, 45316753, 74986921, 123748430, 203686778, 334421510, 547735241

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Number of strict multiset partitions of integer partitions of n. Weigh transform of A000041. - Gus Wiseman, Oct 11 2018

			From _Gus Wiseman_, Oct 11 2018: (Start)
The a(1) = 1 through a(5) = 19 strict multiset partitions:
  {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
         {{1,1}}  {{1,2}}      {{1,3}}        {{1,4}}
                  {{1,1,1}}    {{2,2}}        {{2,3}}
                  {{1},{2}}    {{1,1,2}}      {{1,1,3}}
                  {{1},{1,1}}  {{1},{3}}      {{1,2,2}}
                               {{1,1,1,1}}    {{1},{4}}
                               {{1},{1,2}}    {{2},{3}}
                               {{2},{1,1}}    {{1,1,1,2}}
                               {{1},{1,1,1}}  {{1},{1,3}}
                                              {{1},{2,2}}
                                              {{2},{1,2}}
                                              {{3},{1,1}}
                                              {{1,1,1,1,1}}
                                              {{1},{1,1,2}}
                                              {{1,1},{1,2}}
                                              {{2},{1,1,1}}
                                              {{1},{1,1,1,1}}
                                              {{1,1},{1,1,1}}
                                              {{1},{2},{1,1}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).

Cf. A000041, A001970, A026007, A027998, A248882, A102866, A256142.
Cf. A047968, A050342, A089259, A305551, A320328, A320330, A320331.
Row sums of A360742.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 08 2015

nmax=40; CoefficientList[Series[Product[(1+x^k)^PartitionsP[k],{k,1,nmax}],{x,0,nmax}],x]

A304961 Expansion of Product_{k>=1} (1 + 2^(k-1)*x^k).

Original entry on oeis.org

1, 1, 2, 6, 12, 32, 72, 176, 384, 960, 2112, 4992, 11264, 26112, 58368, 136192, 301056, 688128, 1548288, 3489792, 7766016, 17596416, 38993920, 87293952, 194248704, 432537600, 957349888, 2132803584, 4699717632, 10406068224, 23001563136, 50683969536, 111434268672, 245819768832

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Number of compositions of partitions of n into distinct parts. a(3) = 6: 3, 21, 12, 111, 2|1, 11|1. - Alois P. Heinz, Sep 16 2019
Also the number of ways to split a composition of n into contiguous subsequences with strictly decreasing sums. - Gus Wiseman, Jul 13 2020
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = (-1) * 2^(n-1). - Seiichi Manyama, Aug 22 2020

			From _Gus Wiseman_, Jul 13 2020: (Start)
The a(0) = 1 through a(4) = 12 splittings:
  ()  (1)  (2)    (3)        (4)
           (1,1)  (1,2)      (1,3)
                  (2,1)      (2,2)
                  (1,1,1)    (3,1)
                  (2),(1)    (1,1,2)
                  (1,1),(1)  (1,2,1)
                             (2,1,1)
                             (3),(1)
                             (1,1,1,1)
                             (1,2),(1)
                             (2,1),(1)
                             (1,1,1),(1)
(End)

Cf. A000009, A011782, A022629, A098407, A102866, A266964, A271619, A279785.
The non-strict version is A075900.
Starting with a reversed partition gives A323583.
Starting with a partition gives A336134.
Partitions of partitions are A001970.
Splittings with equal sums are A074854.
Splittings of compositions are A133494.
Splittings with distinct sums are A336127.
Cf. A006951, A063834, A316245, A317715, A319794, A323582, A336135, A336136.

nmax = 33; CoefficientList[Series[Product[(1 + 2^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]

N=40; x='x+O('x^N); Vec(prod(k=1, N, 1+2^(k-1)*x^k)) \\ Seiichi Manyama, Aug 22 2020

G.f.: Product_{k>=1} (1 + A011782(k)*x^k).

a(n) ~ 2^n * exp(2*sqrt(-polylog(2, -1/2)*n)) * (-polylog(2, -1/2))^(1/4) / (sqrt(6*Pi) * n^(3/4)). - Vaclav Kotesovec, Sep 19 2019

A098407 Number of different hierarchical orderings that can be formed from n unlabeled elements with no repetition of subhierarchies.

Original entry on oeis.org

1, 1, 2, 6, 13, 33, 78, 186, 436, 1028, 2394, 5566, 12877, 29689, 68198, 156194, 356599, 811959, 1843956, 4177436, 9442166, 21295934, 47932572, 107677140, 241443980, 540441068, 1207689636, 2694452060, 6002389882, 13351958546, 29659179804, 65794744420, 145768641091

Offset: 0

Thomas Wieder, Sep 07 2004; corrected Sep 09 2004

nonn

a(n) is the number of finite sets of compositions with total sum n. The case of constant sums is A358904, cf. A074854. The case of distinct sums is A304961, ordered A336127. The ordered version (sequences of distinct compositions) is A358907. - Gus Wiseman, Dec 12 2022

			Let a pair of parentheses () indicate a subhierarchy and let square brackets [] denote a set of subhierarchies, that is, a hierarchy (also called a society). Let the ranks be ordered from left to right and separated by a colon; e.g., (2:3) is a subhierarchy with three elements ("individuals") on top and two elements on the bottom rank.
Then the hierarchical ordering for n = 4 is composed of the following sets: [(1:1),(2)]; [(1),(3)]; [(1),(1:1:1)]; [(1),(2:1)]; [(1),(1:2)]; [(4)]; [(2:2)]; [(1:3)]; [(3:1)]; [(1:1:2)]; [(1:2:1)]; [(2:1:1)]; [(1:1:1:1)]; thus a(4) = 13.
For example, the following hierarchy is not allowed: [(1),(1),(1),(1)] because of the repetition of (1).

Alois P. Heinz, Table of n, a(n) for n = 0..3217
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Cf. A011782, A075729.
A034691 counts multisets of compositions, ordered A133494.
A261049 counts sets of partitions, ordered A358906.
Cf. A000009, A000219, A001970, A055887, A075900, A218482, A296122, A304961.

main := proc(n::integer) local a, ListOfPartitions, NumberOfPartitions, APartition, APart, ASet, MultipliticityOfAPart, ndxprttn, ndxprt, Term, Produkt; with(combinat): with(ListTools): a := 0; ListOfPartitions := partition(n); NumberOfPartitions := nops(ListOfPartitions); for ndxprttn from 1 to NumberOfPartitions do APartition := ListOfPartitions[ndxprttn]; ASet := convert(APartition,set); Produkt := 1; for ndxprt from 1 to nops(ASet) do APart := op(ndxprt,ASet); MultipliticityOfAPart := Occurrences(APart, APartition); Term := 2^(APart-1); Term := binomial(Term,MultipliticityOfAPart); Produkt := Produkt * Term; # End of do-loop *** ndxprt ***. end do; a := a + Produkt; # End of do-loop *** ndxprttn ***. end do; print("n, a(n):",n,a); end proc;
PartitionList := proc (n, k) # Authors: # Herbert S. Wilf and Joanna Nordlicht, # Source: # Lecture Notes "East Side West Side,..." # University of Pennsylvania, USA, 2002. # Available from http://www.cis.upenn.edu/~wilf/lecnotes.html # Berechnet die Partitionen von n mit k Summanden. local East, West; if n < 1 or k < 1 or n < k then RETURN([]) elif n = 1 then RETURN([[1]]) else if n < 2 or k < 2 or n < k then West := [] else West := map(proc (x) options operator, arrow; [op(x), 1] end proc, PartitionList(n-1, k-1)) end if; if k <= n-k then East := map(proc(y) options operator, arrow; map(proc (x) options operator, arrow; x+1 end proc, y) end proc, PartitionList(n-k, k)) else East := [] end if; RETURN([op(West), op(East)]) end if end proc;
# second Maple program:
series(exp(add((-1)^(j-1)/j*z^j/(1-2*z^j), j=1..40)), z, 40); # Cf. A102866; Vladeta Jovovic, Feb 19 2008
# alternative Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, `if`(n>1, 0, 1),
      add(b(n-i*j, i-1)*binomial(2^(i-1), j), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, May 22 2018

terms = 32; CoefficientList[Product[(1 + x^k)^(2^(k-1)), {k, 1, terms+1}] + O[x]^(terms+1), x] // Rest (* Jean-François Alcover, Nov 10 2017, after Vladeta Jovovic *)
nmax = 40; CoefficientList[Series[Exp[Sum[-(-1)^k*x^k/(k*(1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 08 2018 *)

a(n) = Sum_{ partitions n = s_1 + ... + s_n } Product_{ Set{s_i} } C(2^(s_i - 1), m(s_i)), where the sum runs over all partitions of n, the product runs over the set of parts of a given partition, s_i is the i-th part in the set of parts, C(k, l) denotes the binomial coefficient and m(s_i) is the multiplicity of part s_i in the given partition.
G.f.: Product_{k>=1} (1+x^k)^(2^(k-1)). - Vladeta Jovovic, Feb 19 2008
a(n) ~ 2^n * exp(sqrt(2*n) - 1/4 + c) / (sqrt(2*Pi) * 2^(3/4) * n^(3/4)), where c = Sum_{k>=2} -(-1)^k / (k*(2^k-2)) = -0.207530918644117743551169251314627032059... - Vaclav Kotesovec, Jun 08 2018
Weigh transform of A011782. - Alois P. Heinz, Jun 25 2018

More terms from Alois P. Heinz, Apr 21 2012

a(0)=1 prepended by Alois P. Heinz, May 22 2018

A034899 Euler transform of powers of 2 [ 2,4,8,16,... ].

Original entry on oeis.org

1, 2, 7, 20, 59, 162, 449, 1200, 3194, 8348, 21646, 55480, 141152, 356056, 892284, 2221208, 5497945, 13533858, 33151571, 80826748, 196219393, 474425518, 1142758067, 2742784304, 6561052331, 15645062126, 37194451937, 88174252924, 208463595471, 491585775018

Offset: 0

N. J. A. Sloane

nonn

			From _Geoffrey Critzer_, Mar 07 2012: (Start)
Per comment in A102866, a(n) is also the number of multisets of binary words of total length n.
a(2) = 7 because the multisets are {a,a}, {b,b}, {a,b}, {aa}, {ab}, {ba}, {bb};
a(3) = 20 because the multisets are {a,a,a}, {b,b,b}, {a,a,b}, {a,b,b}, {a,aa}, {a,ab}, {a,ba}, {a,bb}, {b,aa}, {b,ab}, {b,ba}, {b,bb}, {aaa}, {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {bbb};
where the words within each multiset are separated by commas. (End)

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..3150 (first 900 terms from Alois P. Heinz)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
G. S. Venkatesh and Kurusch Ebrahimi-Fard, A Formal Power Series Approach to Multiplicative Dynamic Feedback, arXiv:2301.04949 [math.OC], 2023.
Thomas Wieder, Additional comments on this sequence

Cf. A034691, the Euler transform of 1, 2, 4, 8, 16, 32, 64, ...
Cf. A075729, A102866.
Column k=2 of A144074.
Row sums of A055375 and of A209406.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^k)^(2^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018 ~

series(1/product((1-x^(n))^(2^(n)),n=1..20),x=0,12); (Wieder)
# second Maple program:
with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add(add(d*2^d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 02 2011

nn = 20; p = Product[1/(1 - x^i)^(2^i), {i, 1, nn}]; CoefficientList[Series[p, {x, 0, nn}], x] (* Geoffrey Critzer, Mar 07 2012 *)

m=50; x='x+O('x^m); Vec(prod(k=1,m,1/(1-x^k)^(2^k))) \\ G. C. Greubel, Nov 09 2018

G.f.: 1/Product_{n>0} (1-x^n)^(2^n). - Thomas Wieder, Mar 06 2005
a(n) ~ c^2 * 2^(n-1) * exp(2*sqrt(n) - 1/2) / (sqrt(Pi) * n^(3/4)), where c = A247003 = exp( Sum_{k>=2} 1/(k*(2^k-2)) ) = 1.3976490050836502... . - Vaclav Kotesovec, Mar 09 2015
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - 2*x^k))). - Ilya Gutkovskiy, Nov 09 2018

More terms from Thomas Wieder, Mar 06 2005

A292804 Number A(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 5, 2, 0, 1, 4, 12, 16, 2, 0, 1, 5, 22, 55, 42, 3, 0, 1, 6, 35, 132, 225, 116, 4, 0, 1, 7, 51, 260, 729, 927, 310, 5, 0, 1, 8, 70, 452, 1805, 4000, 3729, 816, 6, 0, 1, 9, 92, 721, 3777, 12376, 21488, 14787, 2121, 8, 0

Offset: 0

Alois P. Heinz, Sep 23 2017

			A(2,2) = 5: {aa}, {ab}, {ba}, {bb}, {a,b}.
Square array A(n,k) begins:
  1, 1,   1,     1,      1,      1,       1,       1, ...
  0, 1,   2,     3,      4,      5,       6,       7, ...
  0, 1,   5,    12,     22,     35,      51,      70, ...
  0, 2,  16,    55,    132,    260,     452,     721, ...
  0, 2,  42,   225,    729,   1805,    3777,    7042, ...
  0, 3, 116,   927,   4000,  12376,   31074,   67592, ...
  0, 4, 310,  3729,  21488,  83175,  250735,  636517, ...
  0, 5, 816, 14787, 113760, 550775, 1993176, 5904746, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A000009, A102866, A256142, A292838, A292839, A292840, A292841, A292842, A292843, A292844.
Rows n=0-2 give: A000012, A001477, A000326.
Main diagonal gives A292805.
Cf. A144074, A292795, A319501.

h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
    end:
A:= (n, k)-> h(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

h[n_, i_, k_] := h[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[h[n-i*j, i-1, k]* Binomial[k^i, j], {j, 0, n/i}]]];
A[n_, k_] := h[n, n, k];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)

G.f. of column k: Product_{j>=1} (1+x^j)^(k^j).

A(n,k) = Sum_{i=0..k} C(k,i) * A319501(n,i).

A256142 G.f.: Product_{j>=1} (1+x^j)^(3^j).

Original entry on oeis.org

1, 3, 12, 55, 225, 927, 3729, 14787, 57888, 224220, 860022, 3270744, 12343899, 46264257, 172305837, 638039136, 2350109736, 8613851832, 31428857611, 114187160631, 413222547846, 1489829356657, 5352683946903, 19167988920930, 68427472477338, 243559693397025

Offset: 0

Vaclav Kotesovec, Mar 16 2015

nonn

In general, if g.f. = Product_{j>=1} (1+x^j)^(k^j), then a(n) ~ k^n * exp(2*sqrt(n) - 1/2 - c(k)) / (2 * sqrt(Pi) * n^(3/4)), where c(k) = Sum_{m>=2} (-1)^m/(m*(k^(m-1)-1)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of sequence A034691
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

Cf. A102866, A144067, A144074.

Column k=3 of A292804.

nmax=30; CoefficientList[Series[Product[(1+x^k)^(3^k),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 3^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(3^(m-1)-1)) = 0.215985336303958581708278160877115129... .

A208741 Triangular array read by rows. T(n,k) is the number of sets of exactly k distinct binary words with a total of n letters.

Original entry on oeis.org

2, 4, 1, 8, 8, 16, 22, 4, 32, 64, 20, 64, 156, 84, 6, 128, 384, 264, 40, 256, 888, 784, 189, 4, 512, 2048, 2152, 704, 50, 1024, 4592, 5664, 2384, 272, 1, 2048, 10240, 14368, 7328, 1232, 32, 4096, 22496, 35568, 21382, 4704, 248

Offset: 1

Geoffrey Critzer, Mar 08 2012

Equivalently, T(n,k) is the number of integer partitions of n into distinct parts with two types of 1's, four types of 2's, ... , 2^i types of i's,...; where k is the number of summands (of any type).
Row sums = A102866.
Row lengths increase by 1 at n=A061168(offset).

			T(3,2) = 8 because we have: {a,aa}, {a,ab}, {a,ba}, {a,bb}, {b,aa}, {b,ab}, {b,ba}, {b,bb}; 2 word languages with total length 3.
Triangle T(n,k) begins:
   2;
   4,     1;
   8,     8;
  16,    22,    4;
  32,    64,   20;
  64,   156,   84,   6;
  ...

Alois P. Heinz, Rows n = 0..300, flattened
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 64

Cf. A102866, A209406, A360634.

h:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(2^i, j)*x^j, j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(h(n$2)):
seq(T(n), n=1..15);  # Alois P. Heinz, Sep 24 2017

nn=12; p=Product[(1+y x^i)^(2^i), {i,1,nn}]; f[list_] := Select[list, #>0&]; Map[f, Drop[CoefficientList[Series[p[x,y], {x,0,nn}], {x,y}], 1]]//Flatten

O.g.f.: Product_{i>=1} (1 + y*x^i)^(2^i).

A261050 Expansion of Product_{k>=1} (1+x^k)^(Fibonacci(k)).

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 19, 36, 67, 127, 236, 438, 811, 1496, 2750, 5046, 9224, 16827, 30630, 55623, 100803, 182342, 329205, 593326, 1067591, 1917885, 3440207, 6162004, 11021921, 19688757, 35126020, 62590629, 111398910, 198044551, 351700332, 623918086, 1105715149

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015

Cf. A000045, A166861, A261051, A026007, A027998, A248882, A102866, A256142, A260916.

f:= proc(n) option remember; (<<1|1>, <1|0>>^n)[1, 2] end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(binomial(f(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 08 2015

nmax=40; CoefficientList[Series[Product[(1+x^k)^Fibonacci[k],{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ phi^n / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(-1/10 + 2*5^(-1/4)*sqrt(n) + s), where s = Sum_{k>=2} (-1)^(k+1) * phi^k / ((phi^(2*k) - phi^k - 1)*k) = -0.3237251774053525012502809827680337358578568068831886835557918847... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k - x^(2*k)))). - Ilya Gutkovskiy, May 29 2018

A261519 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(2^k).

Original entry on oeis.org

1, 4, 16, 60, 208, 692, 2224, 6940, 21152, 63188, 185488, 536268, 1529648, 4310804, 12017264, 33171916, 90745472, 246201412, 662897232, 1772295020, 4707336848, 12426673188, 32617079280, 85152717404, 221183486496, 571784014244, 1471463190032, 3770577250716

Offset: 0

Vaclav Kotesovec, Aug 23 2015

nonn

Convolution of A034899 and A102866.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO]
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

Cf. A156616, A261520, A260916, A001934, A015128.

nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(2^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 2^n * exp(2*sqrt(2*n) - 1 + c) / (sqrt(Pi) * 2^(3/4) * n^(3/4)), where c = 2 * Sum_{j>=1} 1/((2*j+1)*(2^(2*j)-1)) = 0.2545212486386431009939814261118792033...

A306156 Inverse Weigh transform of 2^n.

Original entry on oeis.org

2, 3, 2, 6, 6, 11, 18, 36, 56, 105, 186, 346, 630, 1179, 2182, 4116, 7710, 14588, 27594, 52482, 99858, 190743, 364722, 699216, 1342176, 2581425, 4971008, 9587574, 18512790, 35792449, 69273666, 134219796, 260300986, 505294125, 981706806, 1908881548, 3714566310

Offset: 1

Seiichi Manyama, Jun 23 2018

nonn

			(1+x)^2*(1+x^2)^3*(1+x^3)^2*(1+x^4)^6* ... = 1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + ... .

Seiichi Manyama, Table of n, a(n) for n = 1..3000
Christian G. Bower, PARI programs for transforms, 2007.
N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Inverse Weigh transform of b^n: this sequence (b=2), A306157 (b=3), A306158 (b=4), A306159 (b=5).

Cf. A000079, A038067, A102866.

Product_{k>=1} (1+x^k)^a(k) = 1/(1-2x).

a(n) = (1/n) * (2^n + Sum_{d

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A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.

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A304961 Expansion of Product_{k>=1} (1 + 2^(k-1)*x^k).

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A098407 Number of different hierarchical orderings that can be formed from n unlabeled elements with no repetition of subhierarchies.

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A034899 Euler transform of powers of 2 [ 2,4,8,16,... ].

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A292804 Number A(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A256142 G.f.: Product_{j>=1} (1+x^j)^(3^j).

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A208741 Triangular array read by rows. T(n,k) is the number of sets of exactly k distinct binary words with a total of n letters.

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