A358369 -id:A358369 - 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

A092119 EULER transform of A001511.

Original entry on oeis.org

1, 1, 3, 4, 10, 13, 26, 35, 66, 88, 150, 202, 331, 442, 688, 919, 1394, 1848, 2716, 3590, 5174, 6796, 9589, 12542, 17440, 22680, 31055, 40208, 54420, 70096, 93772, 120256, 159380, 203436, 267142, 339573, 442478, 560050, 724302, 913198, 1173375, 1473622

Offset: 0

Vladeta Jovovic, Mar 29 2004

nonn

From Gary W. Adamson, Feb 11 2010: (Start)
Given A000041, P(x) = A(x)/A(x^2) with P(x) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...),
A(x) = (1 + x + 3x^2 + 4x^3 + 10x^4 + 13x^5 + ...),
A(x^2) = (1 + x^2 + 3x^4 + 4x^6 + 10x^8 + ...), where A092119 = (1, 1, 3, 4, 10, ...) = Euler transform of the ruler sequence, A001511. (End)
Let M = triangle A173238 as an infinite lower triangular matrix. Then A092119 = lim_{n->infinity} M^n. Let P(x) = polcoeff A000041 = (1 + x + 2x^2 + 3x^3 + ...), and A(x) = polcoeff A092119. Then P(x) = A(x) / A(x^2), an example of a conjectured infinite set of operations (cf. A173238). - Gary W. Adamson, Feb 13 2010

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
N. J. A. Sloane, Transforms

Cf. A000041. - Gary W. Adamson, Feb 11 2010

Cf. A173241.

# Uses EulerTransform from A358369.
t := EulerTransform(n -> padic[ordp](2*n, 2)):
seq(t(n), n = 0..41); # Peter Luschny, Nov 18 2022

m = 42;
1/Product[QPochhammer[x^(2^k)], {k, 0, Log[2, m]//Ceiling}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Jan 14 2020, after Joerg Arndt *)

N=66; x='x+O('x^N); /* that many terms */
gf=1/prod(e=0,ceil(log(N)/log(2)),eta(x^(2^e)));
Vec(gf) /* show terms */ /* Joerg Arndt, Jun 21 2011 */

G.f.: 1/Product_{k>=0} P(x^(2^k)) where P(x) = Product_{k>=1} (1 - x^k). - Joerg Arndt, Jun 21 2011

A219224 G.f.: exp( Sum_{n>=1} A005063(n)*x^n/n ), where A005063(n) = sum of squares of primes dividing n.

Original entry on oeis.org

1, 0, 2, 3, 3, 11, 10, 26, 32, 51, 90, 117, 198, 283, 417, 610, 890, 1284, 1848, 2615, 3716, 5217, 7289, 10222, 14158, 19514, 26882, 36805, 50131, 68428, 92466, 125128, 168093, 225775, 302171, 402876, 536730, 711601, 942009, 1243513, 1638395, 2152828, 2823004

Offset: 0

Paul D. Hanna, Nov 15 2012

nonn

Euler transform of A061397. - Peter Luschny, Nov 21 2022

			G.f.: A(x) = 1 + 2*x^2 + 3*x^3 + 3*x^4 + 11*x^5 + 10*x^6 + 26*x^7 + 32*x^8 +...
where
log(A(x)) = 4*x^2/2 + 9*x^3/3 + 4*x^4/4 + 25*x^5/5 + 13*x^6/6 + 49*x^7/7 + 4*x^8/8 + 9*x^9/9 + 29*x^10/10 + 121*x^11/11 + 13*x^12/12 + 169*x^13/13 + 53*x^14/14 + 34*x^15/15 +...+ A005063(n)*x^n/n +...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A000607, A005063, A220427, A061397.

# The function EulerTransform is defined in A358369.
a := EulerTransform(n -> ifelse(isprime(n), n, 0)):
seq(a(n), n = 0..42); # Peter Luschny, Nov 21 2022

a[n_] := SeriesCoefficient[ Exp[ Sum[ DivisorSum[k, Boole[PrimeQ[#]] * #^2&] * x^k/k, {k, 1, n+1}]], {x, 0, n}]; Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Jul 11 2017, from PARI *)

{a(n)=polcoeff(exp(sum(k=1,n+1,sumdiv(k,d,isprime(d)*d^2)*x^k/k)+x*O(x^n)),n)}
for(n=0,50,print1(a(n),", "))

A111335 Let qf(a,q) = Product_{j>=0} (1 - a*q^j); g.f. is qf(q^3,q^4)/qf(q,q^4).

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 1, 0, -1, 0, 2, 1, -1, -1, 1, 2, -1, -2, 1, 3, 1, -3, -1, 3, 1, -3, -2, 4, 4, -3, -4, 3, 5, -3, -7, 2, 9, 0, -9, -1, 10, 3, -11, -5, 12, 8, -11, -10, 10, 12, -11, -15, 11, 19, -7, -21, 6, 24, -5, -28, 1, 31, 4, -33, -8, 36, 12, -38, -18, 40, 27, -40, -33, 40, 39, -40, -49, 38, 60, -34, -67, 30, 75, -25, -87, 18, 98

Offset: 0

N. J. A. Sloane, Nov 09 2005

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A111330.

# Uses EulerTransform from A358369.
a := EulerTransform(BinaryRecurrenceSequence(0, -1)):
seq(a(n), n=0..86); # Peter Luschny, Nov 17 2022

{a(n)=if(n<0, 0, polcoeff( prod(k=0,n\2, (1-x^(2*k+1))^(-(-1)^k), 1+x*O(x^n)), n))} /* Michael Somos, Nov 11 2005 */

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, -1)
a = EulerTransform(b)
print([a(n) for n in range(87)]) # Peter Luschny, Nov 17 2022

Euler transform of period 4 sequence [1, 0, -1, 0, ...]. - Michael Somos, Nov 10 2005
G.f.: Product_{k>0} (1-x^(2k-1))^((-1)^k). - Michael Somos, Nov 11 2005
G.f.: exp( Sum_{k >= 1} 1/(k*(x^k + x^(-k))) ). - Peter Bala, Sep 28 2023

A117410 Expansion of q^(-5/24) * eta(q^2)^3 / eta(q) in powers of q.

Original entry on oeis.org

1, 1, -1, 0, -1, -2, 1, -1, -1, 0, 1, 1, -1, 1, 0, 2, 1, 0, 0, -1, 2, 1, 0, -1, 0, -1, 0, -1, 1, 1, -3, 0, -1, -1, -1, 1, 0, 0, 0, -1, -2, 0, 1, 0, 1, 0, 1, 0, 0, -1, 2, -1, 0, 1, 1, 3, 0, -1, 0, 1, -1, 0, 1, 0, 0, 2, 0, 1, -1, 0, -2, -1, 1, 0, 0, -1, 0, 0, 1, -1, 0, -1, -1, -1, 0, -2, -1, 0, 2, 1, -2, 0, 1, -1, 0, -2, -1, 1, -1, 1, 0, 0, 0, 1, 0

Offset: 0

Michael Somos, Mar 13 2006

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x - x^2 - x^4 - 2*x^5 + x^6 - x^7 - x^8 + x^10 + x^11 - x^12 + x^13 + ...
G.f. = q^5 + q^29 - q^53 - q^101 - 2*q^125 + q^149 - q^173 - q^197 + q^245 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. Gordon and D. Sinor, Multiplicative properties of eta-products, Number theory, Madras 1987, pp. 173-200, Lecture Notes in Math., 1395, Springer, Berlin, 1989. see page 183. MR1019331 (90k:11050).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A107034.

# Uses EulerTransform from A358369.
a := EulerTransform(BinaryRecurrenceSequence(0, 1, -2)):
seq(a(n), n = 0..104); # Peter Luschny, Nov 17 2022

a[ n_] := SeriesCoefficient[ QPochhammer[x^2]^3 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / eta(x + A), n))};

q='q+O('q^99); Vec(eta(q^2)^3/eta(q)) \\ Altug Alkan, Apr 17 2018

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, -2)
a = EulerTransform(b)
print([a(n) for n in range(105)]) # Peter Luschny, Nov 17 2022

Expansion of psi(x)^2 * chi(-x) = f(-x)^2 / chi(-x)^3 = f(-x)^5 / phi(-x)^3 = f(-x^2)^2 / chi(-x) = f(-x^2)^3 / f(-x) = psi(x) * f(-x^2) = f(x) * f(-x^4) = phi(-x)^2 / chi(-x)^5 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jan 31 2015
Euler transform of period 2 sequence [ 1, -2, ...].
Given A = A0 + A1 + A2 + A3 + A4 is the 5-section, then 0 = A3 * A1^2 - A2 * A4^2.
Given A = A0 + A1 + A2 + A3 + A4 + A5 + A6 is the 7-section, then 0 = A0*A6 + A1*A5 + A2*A4 + 4*A3^2, A3 = x^10 * A(x^49).
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2.
A107034(n) = (-1)^n * a(n).

cp's OEIS Frontend

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

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A092119 EULER transform of A001511.

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A219224 G.f.: exp( Sum_{n>=1} A005063(n)*x^n/n ), where A005063(n) = sum of squares of primes dividing n.

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A111335 Let qf(a,q) = Product_{j>=0} (1 - a*q^j); g.f. is qf(q^3,q^4)/qf(q,q^4).

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A117410 Expansion of q^(-5/24) * eta(q^2)^3 / eta(q) in powers of q.

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A358449 Euler transform of (0, 1, -2, 4, -8, 16, ...), (cf. A122803).

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