A034899 -id:A034899 - cp's OEIS frontend

A034691 Euler transform of powers of 2 [1,2,4,8,16,...].

1, 1, 3, 7, 18, 42, 104, 244, 585, 1373, 3233, 7533, 17547, 40591, 93711, 215379, 493735, 1127979, 2570519, 5841443, 13243599, 29953851, 67604035, 152258271, 342253980, 767895424, 1719854346, 3845443858

Offset: 0

N. J. A. Sloane

nonn

This is the number of different hierarchical orderings that can be formed from n unlabeled elements: these are divided into groups and the elements in each group are then arranged in an "unlabeled preferential arrangement" or "composition" as in A000079. - Thomas Wieder and N. J. A. Sloane, Jun 10 2003
From Gus Wiseman, Mar 03 2016: (Start)
The original Sloane-Wieder definition, "To obtain a hierarchical ordering we partition the elements into unlabeled nonempty subsets and form a composition of each subset," [arXiv:math/0307064] has two possible meanings. The first possible meaning is that we should (1) choose a set partition pi of {1...n} and (2) for each block of pi choose a composition of the number of elements. In this case the correct number of such structures would evidently be counted by A004211 which differs from a(n) for n > 2.
The other possible meaning is that after we have done (1) and (2) above we (3) "forget" the choice of pi. We will have produced a collection M of multisets of compositions. The span of M (its set of distinct elements) is correctly counted by A034691 and it seems that non-isomorphic hierarchical orderings of unlabeled sets are nothing more than multisets of compositions. This discovery is due to Wieder. (End)
The asymptotic formula in the article by N. J. A. Sloane and Thomas Wieder, "The Number of Hierarchical Orderings" (Theorem 3) is incorrect (a multiplicative factor of 1.397... is missing, see my formula below). - Vaclav Kotesovec, Sep 08 2014
Number of partitions of n into 1 sort of 1, 2 sorts of 2's, 4 sorts of 3's, ..., 2^(k-1) sorts of k's, ... . - Joerg Arndt, Sep 09 2014
Also number of normal multiset partitions of weight n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Mar 03 2016

			The normal multiset partitions for a(4) = 18: {{1111},{1222},{1122},{1112},{1233},{1223},{1123},{1234},{1,111},{1,122},{1,112},{1,123},{11,11},{11,12},{12,12},{1,1,11},{1,1,12},{1,1,1,1}}

Vaclav Kotesovec, Table of n, a(n) for n = 0..3190 (first 300 terms from T. D. Noe)
Vaclav Kotesovec, Asymptotics of sequence A034691.
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
Thomas Wieder, An explicit formula for the n-th term.
Thomas Wieder, The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [From _Thomas Wieder_, Nov 14 2009]

Cf. A034899, A075729, A247003, A004211, A104500 (Euler transform), A290222 (Multiset transform).

oo := 101: mul( 1/(1-x^j)^(2^(j-1)),j=1..oo): series(%,x,oo): t1 := seriestolist(%); A034691 := n-> t1[n+1];
with(combstruct); SetSeqSetU := [T, {T=Set(S), S=Sequence(U,card >= 1), U=Set(Z,card >=1)},unlabeled]; seq(count(SetSeqSetU,size=j),j=1..12);
# Alternative, uses EulerTransform from A358369:
a := EulerTransform(BinaryRecurrenceSequence(2, 0)):
seq(a(n), n = 0..27); # Peter Luschny, Nov 17 2022

nn = 30; b = Table[2^n, {n, 0, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}],  x] (* T. D. Noe, Nov 21 2011 *)
Table[SeriesCoefficient[E^(Sum[x^k/(1 - 2*x^k)/k, {k, 1, n}]), {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 08 2014 *)
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
allnmsp[0]={};allnmsp[1]={{{1}}};allnmsp[n_Integer]:=allnmsp[n]=Join[allnmsp[n-1],List/@allnorm[n],Join@@Function[ptn,Append[ptn,#]&/@Select[allnorm[n-Length[Join@@ptn]],OrderedQ[{Last[ptn],#}]&]]/@allnmsp[n-1]];
Apply[SequenceForm,Select[allnmsp[4],Length[Join@@#]===4&],{2}] (* to construct the example *)
Table[Length[Complement[allnmsp[n],allnmsp[n-1]]],{n,1,8}] (* Gus Wiseman, Mar 03 2016 *)

A034691(n,l=1+O('x^(n+1)))={polcoeff(1/prod(k=1,n,(l-'x^k)^2^(k-1)),n)} \\ Michael Somos, Nov 21 2011, edited by M. F. Hasler, Jul 24 2017

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(2, 0)
b = EulerTransform(a)
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

G.f.: 1 / Product_{n>=1} (1-x^n)^(2^(n-1)).
Recurrence: a(n) = (1/n) * Sum_{m=1..n} a(n-m)*c(m) where c(m) = A083413(m).
a(n) ~ c * 2^n * exp(sqrt(2*n)) / (sqrt(2*Pi) * exp(1/4) * 2^(3/4) * n^(3/4)), where c = exp( Sum_{k>=2} 1/(k*(2^k-2)) ) = 1.3976490050836502... (see A247003). - Vaclav Kotesovec, Sep 08 2014

A102866 Number of finite languages over a binary alphabet (set of nonempty binary words of total length n).

Original entry on oeis.org

1, 2, 5, 16, 42, 116, 310, 816, 2121, 5466, 13937, 35248, 88494, 220644, 546778, 1347344, 3302780, 8057344, 19568892, 47329264, 114025786, 273709732, 654765342, 1561257968, 3711373005, 8797021714, 20794198581, 49024480880, 115292809910, 270495295636

Offset: 0

Philippe Flajolet, Mar 01 2005

nonn

Analogous to A034899 (which also enumerates multisets of words)

			a(2) = 5 because the sets are {a,b}, {aa}, {ab}, {ba}, {bb}.
a(3) = 16 because the sets are {a,aa}, {a,ab}, {a,ba}, {a,bb}, {b,aa}, {b,ab}, {b,ba}, {b,bb}, {aaa}, {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {bbb}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 64
Stefan Gerhold, Counting finite languages by total word length, INTEGERS 11 (2011), #A44.
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. A000079, A034899, A256142.
Column k=2 of A292804.
Row sums of A208741 and of A360634.

series(exp(add((-1)^(j-1)/j*(2*z^j)/(1-2*z^j),j=1..40)),z,40);

nn = 20; p = Product[(1 + x^i)^(2^i), {i, 1, nn}]; CoefficientList[Series[p, {x, 0, nn}], x] (* Geoffrey Critzer, Mar 07 2012 *)
CoefficientList[Series[E^Sum[(-1)^(k-1)/k*(2*x^k)/(1-2*x^k), {k,1,30}], {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 13 2014 *)

G.f.: exp(Sum((-1)^(j-1)/j*(2*z^j)/(1-2*z^j), j=1..infinity)).
Asymptotics (Gerhold, 2011): a(n) ~ c * 2^(n-1)*exp(2*sqrt(n)-1/2) / (sqrt(Pi) * n^(3/4)), where c = exp( Sum_{k>=2} (-1)^(k-1)/(k*(2^(k-1)-1)) ) = 0.6602994483152065685... . - Vaclav Kotesovec, Sep 13 2014
Weigh transform of A000079. - Alois P. Heinz, Jun 25 2018

A144074 Number A(n,k) of multisets 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, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 20, 5, 0, 1, 5, 26, 64, 59, 7, 0, 1, 6, 40, 148, 276, 162, 11, 0, 1, 7, 57, 285, 843, 1137, 449, 15, 0, 1, 8, 77, 488, 2020, 4632, 4648, 1200, 22, 0, 1, 9, 100, 770, 4140, 13876, 25124, 18585, 3194, 30, 0, 1, 10, 126

Offset: 0

Alois P. Heinz, Sep 09 2008

Column k > 1 is asymptotic to k^n * exp(2*sqrt(n) - 1/2 + c(k)) / (2 * sqrt(Pi) * n^(3/4)), where c(k) = Sum_{m>=2} 1/(m*(k^(m-1)-1)). - Vaclav Kotesovec, Mar 14 2015

			A(4,1) = 5: {aaaa}, {aaa,a}, {aa,aa}, {aa,a,a}, {a,a,a,a}.
A(2,2) = 7: {aa}, {a,a}, {bb}, {b,b}, {ab}, {ba}, {a,b}.
A(2,3) = 15: {aa}, {a,a}, {bb}, {b,b}, {cc}, {c,c}, {ab}, {ba}, {a,b}, {ac}, {ca}, {a,c}, {bc}, {cb}, {b,c}.
A(3,2) = 20: {aaa}, {a,aa}, {a,a,a}, {bbb}, {b,bb}, {b,b,b}, {aab}, {aba}, {baa}, {a,ab}, {a,ba}, {aa,b}, {a,a,b}, {bba}, {bab}, {abb}, {b,ba}, {b,ab}, {bb,a}, {b,b,a}.
Square array begins:
  1, 1,   1,    1,    1,     1, ...
  0, 1,   2,    3,    4,     5, ...
  0, 2,   7,   15,   26,    40, ...
  0, 3,  20,   64,  148,   285, ...
  0, 5,  59,  276,  843,  2020, ...
  0, 7, 162, 1137, 4632, 13876, ...

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

Columns k=0-10 give: A000007, A000041, A034899, A144067, A144068, A144069, A144070, A144071, A144072, A144073, A292837.
Rows n=0-2 give: A000012, A001477, A005449.
Main diagonal gives A252654.
Cf. A004248, A257740, A256142, A292804.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->k^j)(n); seq(seq(A(n, d-n), n=0..d), d=0..14);

a[n_, k_] := SeriesCoefficient[ Product[1/(1-x^j)^(k^j), {j, 1, n}], {x, 0, n}]; a[0, ] = 1; a[?Positive, 0] = 0;
Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 15 2014 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d p[d], {d, Divisors[j]}] b[n-j], {j, 1, n}]/n]; b];
A[n_, k_] := etr[k^#&][n];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 30 2020, after Alois P. Heinz *)

G.f. of column k: Product_{j>=1} 1/(1-x^j)^(k^j).
Column k is Euler transform of the powers of k.
T(n,k) = Sum_{i=0..k} C(k,i) * A257740(n,k-i). - Alois P. Heinz, May 08 2015

Name changed by Alois P. Heinz, Sep 21 2018

A209406 Triangular array read by rows: T(n,k) is the number of multisets of exactly k nonempty binary words with a total of n letters.

Original entry on oeis.org

2, 4, 3, 8, 8, 4, 16, 26, 12, 5, 32, 64, 44, 16, 6, 64, 164, 132, 62, 20, 7, 128, 384, 376, 200, 80, 24, 8, 256, 904, 1008, 623, 268, 98, 28, 9, 512, 2048, 2632, 1792, 870, 336, 116, 32, 10, 1024, 4624, 6624, 5040, 2632, 1117, 404, 134, 36, 11

Offset: 1

Geoffrey Critzer, Mar 08 2012

Equivalently, T(n,k) is the number of partitions of the integer n with two types of 1's, four types of 2's, ..., 2^i types of i's...; having exactly k summands (of any type).

Row sums = A034899.

			Triangle T(n,k) begins:
    2;
    4,    3;
    8,    8,    4;
   16,   26,   12,    5;
   32,   64,   44,   16,   6;
   64,  164,  132,   62,  20,   7;
  128,  384,  376,  200,  80,  24,   8;
  256,  904, 1008,  623, 268,  98,  28,  9;
  512, 2048, 2632, 1792, 870, 336, 116, 32, 10;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Index entries for triangles generated by the Multiset Transformation

Cf. A034899, A055375, A208741, A290222, A292506.

T(2n,n) gives A359962.

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
       binomial(2^i+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Apr 13 2017

nn = 10; p[x_, y_] := Product[1/(1 - y x^i)^(2^i), {i, 1, nn}]; f[list_] := Select[lst, # > 0 &]; Map[f, Drop[CoefficientList[Series[p[x, y], {x, 0, nn}], {x, y}], 1]] // Flatten

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

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...

A297567 Number of nonisomorphic proper colorings of partition star graph using three colors.

Original entry on oeis.org

3, 6, 9, 12, 12, 24, 24, 15, 36, 30, 48, 48, 18, 48, 60, 72, 96, 96, 96, 21, 60, 90, 60, 96, 192, 108, 144, 192, 192, 192, 24, 72, 120, 120, 120, 288, 240, 216, 192, 384, 384, 288, 384, 384, 384, 27, 84, 150, 180, 105, 144, 384, 480, 324, 432, 240, 576, 480, 768, 408, 384, 768, 768, 576, 768, 768, 768, 30, 96, 180, 240, 210, 168, 480, 720, 480, 432, 864, 360, 288, 768, 960, 1152, 1536, 816, 480, 1152, 960, 1536, 1536, 768

Offset: 0

Marko Riedel, Dec 31 2017

A partition star graph consists of a multiset of paths with lengths given by the elements of the partition attached to a distinguished root node. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the star graph corresponding to the partition.

			Rows are:
   3;
   6;
   9, 12;
  12, 24, 24;
  15, 36, 30, 48, 48;
  18, 48, 60, 72, 96, 96, 96;

Marko Riedel et al., Orbital chromatic polynomials
Marko Riedel, Maple code computing OCP for sequences A297567, A297568, A297569, A297570.

Cf. A297568, A297569, A297570.
Row sums give 3*A034899.
Row lengths give A000041.

b:= (n, i)-> `if`(n=0, [3], `if`(i<1, [], [seq(map(x-> x*
     binomial(2^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jan 14 2018

b[n_, i_] := If[n == 0, {3}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 2^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
T[n_] :=  b[n, n];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 17 2018, after Alois P. Heinz *)

For a partition lambda we have the OCP: k Product_{p^v in lambda} C((k-1)^p+v-1, v). Here we have k=3.

A055375 Euler transform of Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 7, 7, 3, 5, 14, 21, 14, 5, 7, 26, 48, 48, 26, 7, 11, 45, 103, 131, 103, 45, 11, 15, 75, 198, 312, 312, 198, 75, 15, 22, 120, 366, 674, 830, 674, 366, 120, 22, 30, 187, 637, 1359, 1961, 1961, 1359, 637, 187, 30, 42, 284, 1078, 2584, 4302, 5066, 4302, 2584, 1078, 284, 42

Offset: 0

Christian G. Bower, May 16 2000

Number of partitions of n objects, k of which are black, into parts each of which is a sequence of objects. E.g. T(3,1) = 7; the partitions are [BWW], [WBW], [WWB], [BW,W], [WB,W], [WW,B] and [B,W,W]. - Franklin T. Adams-Watters, Jan 10 2007

			Triangle begins
   1;
   1,  1;
   2,  3,   2;
   3,  7,   7,   3;
   5, 14,  21,  14,   5;
   7, 26,  48,  48,  26,   7;
  11, 45, 103, 131, 103,  45, 11;
  15, 75, 198, 312, 312, 198, 75, 15;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Row sums give A034899.
Columns k=0-1 give: A000041, A014153(n-1) for n>=1.
T(2n,n) gives A360626.
Cf. A007318, A034691, A209406, A360634.

g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= (n, k)-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Feb 14 2023

nmax = 10; pp = Product[Product[1/(1 - x^i*y^j)^Binomial[i, j], {j, 0, i}], {i, 1, nmax}]; t[n_, k_] := SeriesCoefficient[pp, {x, 0, n}, {y, 0, k}]; Table[t[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 18 2017 *)

G.f.: Product_{i>=1} Product_{j=0..i} 1/(1 - x^i y^j)^C(i,j). - Franklin T. Adams-Watters, Jan 10 2007

Sum_{k=0..2n} (-1)^k * T(2n,k) = A034691(n). - Alois P. Heinz, Dec 05 2023

A247003 Decimal expansion of a constant related to A034691.

Original entry on oeis.org

1, 3, 9, 7, 6, 4, 9, 0, 0, 5, 0, 8, 3, 6, 5, 0, 2, 8, 5, 0, 6, 5, 0, 7, 4, 5, 9, 8, 5, 2, 6, 7, 9, 1, 1, 5, 9, 0, 0, 7, 8, 1, 1, 4, 2, 9, 4, 4, 0, 7, 2, 8, 9, 9, 6, 4, 8, 3, 8, 7, 4, 0, 4, 8, 8, 5, 4, 6, 6, 5, 7, 2, 0, 6, 6, 0, 8, 3, 3, 8, 5, 7, 8, 2, 0, 7, 5, 7, 3, 3, 2, 3, 3, 1, 0, 2, 4, 8, 2, 0, 4, 0, 0, 1, 5

Offset: 1

Vaclav Kotesovec, Sep 09 2014

			1.397649005083650285065074598526791159007811429440728996483874...

Vaclav Kotesovec, Asymptotics of sequence A034691

Cf. A034691, A034899, A261043, A304962, A319918.

evalf(exp(sum(1/(k*(2^k-2)), k=2..infinity)), 100)

Equals exp( Sum_{k>=2} 1/(k*(2^k-2)) ).

A290222 Multiset transform of A011782, powers of 2: 1, 2, 4, 8, 16, ...

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 8, 7, 2, 1, 0, 16, 16, 7, 2, 1, 0, 32, 42, 20, 7, 2, 1, 0, 64, 96, 54, 20, 7, 2, 1, 0, 128, 228, 140, 59, 20, 7, 2, 1, 0, 256, 512, 360, 156, 59, 20, 7, 2, 1, 0, 512, 1160, 888, 422, 162, 59, 20, 7, 2, 1, 0, 1024, 2560, 2168, 1088, 442, 162, 59, 20, 7, 2, 1

Offset: 0

M. F. Hasler, Jul 24 2017

T(n,k) is the number of multisets of exactly k binary words with a total of n letters and each word beginning with 1. T(4,2) = 7: {1,100}, {1,101}, {1,110}, {1,111}, {10,10}, {10,11}, {11,11}. - Alois P. Heinz, Sep 18 2017

			The triangle starts:
1;
0    1;
0    2    1;
0    4    2    1;
0    8    7    2    1;
0   16   16    7    2   1;
0   32   42   20    7   2   1;
0   64   96   54   20   7   2  1;
0  128  228  140   59  20   7  2  1;
0  256  512  360  156  59  20  7  2  1;
0  512 1160  888  422 162  59 20  7  2  1;
0 1024 2560 2168 1088 442 162 59 20  7  2  1;
(...)

Alois P. Heinz, Rows n = 0..140, flattened
Index entries for triangles generated by the Multiset Transformation

Cf. A034691 (row sums), A000007 (column k=0), A011782 (column k=1), A178945(n-1) (column k=2).
The reverse of the n-th row converges to A034899.
Cf. A000079, A209406, A292506.

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(binomial(2^(i-1)+j-1, j)*
         b(n-i*j, i-1, p-j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Sep 12 2017

b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[Binomial[2^(i - 1) + j - 1, j] b[n - i j, i - 1, p - j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

G.f.: 1 / Product_{j>=1} (1-y*x^j)^(2^(j-1)). - Alois P. Heinz, Sep 18 2017

A319918 Expansion of Product_{k>=1} 1/(1 - x^k)^(2^k-1).

Original entry on oeis.org

1, 1, 4, 11, 32, 84, 230, 597, 1567, 4020, 10286, 25994, 65387, 163065, 404617, 997687, 2448220, 5977334, 14530835, 35173496, 84814982, 203760809, 487845377, 1164191563, 2769721073, 6570218773, 15542642042, 36671354125, 86306246887, 202637312099, 474684979292, 1109539437382

Offset: 0

Ilya Gutkovskiy, Oct 01 2018

nonn

Convolution of A010815 and A034899.

Euler transform of A000225.

Alois P. Heinz, Table of n, a(n) for n = 0..3171
N. J. A. Sloane, Transforms

Cf. A000225, A007070, A010815, A034691, A034899, A319919.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
       d*(2^d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 13 2021

nmax = 31; CoefficientList[Series[Product[1/(1 - x^k)^(2^k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[Exp[Sum[x^k/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (2^d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]

G.f.: exp(Sum_{k>=1} x^k/(k*(1 - x^k)*(1 - 2*x^k))).

a(n) ~ A247003^2 * exp(2*sqrt(n) - 1/2) * 2^(n-1) / (A065446 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Sep 15 2021

cp's OEIS Frontend

A034691 Euler transform of powers of 2 [1,2,4,8,16,...].

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A102866 Number of finite languages over a binary alphabet (set of nonempty binary words of total length n).

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A144074 Number A(n,k) of multisets 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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A209406 Triangular array read by rows: T(n,k) is the number of multisets of exactly k nonempty binary words with a total of n letters.

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

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A297567 Number of nonisomorphic proper colorings of partition star graph using three colors.

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A055375 Euler transform of Pascal's triangle A007318.

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