A166861 -id:A166861 - cp's OEIS frontend

A030009 Euler transform of primes.

1, 2, 6, 15, 37, 85, 192, 414, 879, 1816, 3694, 7362, 14480, 28037, 53644, 101379, 189587, 350874, 643431, 1169388, 2108045, 3770430, 6694894, 11804968, 20679720, 35999794, 62298755, 107198541, 183462856, 312357002, 529173060, 892216829, 1497454396, 2502190992

Offset: 0

N. J. A. Sloane

nonn

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

Cf. A007441.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*ithprime(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  #  Alois P. Heinz, Sep 06 2008

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*Prime[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 16 2014, after Alois P. Heinz *)

a(n)=if(n<0,0,polcoeff(prod(i=1,n,(1-x^i)^-prime(i),1+x*O(x^n)),n))

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: nth_prime(n))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{n>=1} (1-x^n)^(-prime(n)).

A023880 Number of partitions in expanding space.

Original entry on oeis.org

1, 1, 5, 32, 298, 3531, 51609, 894834, 17980052, 410817517, 10518031721, 298207687029, 9273094072138, 313757506696967, 11474218056441581, 450961669608632160, 18954582520550896213, 848384721904740036422, 40285256621556957160307, 2022695276960566890383148

Offset: 0

Olivier Gérard

nonn

Also partitions of n into 1 sort of 1, 4 sorts of 2, 27 sorts of 3, ..., k^k sorts of k. - Joerg Arndt, Feb 04 2015

Alois P. Heinz, Table of n, a(n) for n = 0..300

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^(k^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(
      add(d*d^d, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 04 2015

nmax=20; CoefficientList[Series[Product[1/(1-x^k)^(k^k),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 14 2015 *)

m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^(k^k))) \\ G. C. Greubel, Oct 31 2018

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n^n)
print([b(n) for n in range(20)]) # Peter Luschny, Nov 11 2020

G.f.: 1 / Product_{k>=1} (1 - x^k)^(k^k).
a(n) ~ n^n * (1 + exp(-1)/n + (exp(-1)/2 + 5*exp(-2))/n^2). - Vaclav Kotesovec, Mar 14 2015
a(n) = (1/n)*Sum_{k=1..n} A283498(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 11 2017

A258349 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)/2).

Original entry on oeis.org

1, 0, 1, 3, 7, 13, 28, 52, 107, 203, 396, 741, 1409, 2596, 4813, 8777, 15972, 28737, 51553, 91644, 162288, 285377, 499653, 869758, 1507615, 2599974, 4465606, 7635607, 13005252, 22061424, 37287395, 62788012, 105365891, 176211393, 293741195, 488101711, 808604106

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000294, A023871, A027999, A258347, A258348, A258350, A258351, A258352.

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

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n,2))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

a(n) ~ 1 / (2^(155/96) * 15^(11/96) * Pi^(1/24) * n^(59/96)) * exp(-Zeta'(-1)/2 - Zeta(3) / (8*Pi^2) - 75*Zeta(3)^3 / (2*Pi^8) - 15^(5/4) * Zeta(3)^2 / (2^(7/4) * Pi^5) * n^(1/4) - sqrt(15/2) * Zeta(3) / Pi^2 * sqrt(n) + 2^(7/4)*Pi / (3*15^(1/4)) * n^(3/4)), where Zeta(3) = A002117, Zeta'(-1) = A084448 = 1/12 - log(A074962).

G.f.: exp(Sum_{k>=1} (sigma_3(k) - sigma_2(k))*x^k/(2*k)). - Ilya Gutkovskiy, Aug 22 2018

A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)/6).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 21, 39, 76, 145, 294, 581, 1169, 2276, 4435, 8494, 16237, 30768, 58221, 109466, 205223, 382658, 710808, 1314091, 2420437, 4439753, 8115645, 14781062, 26833241, 48550863, 87575527, 157480827, 282362462, 504819198, 900058558, 1600424247

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000335, A023872, A258346, A258347, A258348, A258349, A258350, A258351.

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)*(k-2)/6),{k,1,nmax}],{x,0,nmax}],x]

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n, 3))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

a(n) ~ Zeta(5)^(379/3600) / (2^(521/1800) * sqrt(5*Pi) * n^(2179/3600)) * exp(Zeta'(-1)/3 + Zeta(3)/(8*Pi^2) - Pi^16 / (3110400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (216000 * Zeta(5)^2) - Zeta(3)^2/(90*Zeta(5)) + Zeta'(-3)/6 + (-Pi^12 / (10800000 * 2^(2/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (900 * 2^(2/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (36000 * 2^(4/5) * Zeta(5)^(7/5)) + Zeta(3) / (3 * 2^(4/5) * Zeta(5)^(2/5))) * n^(2/5) - Pi^4 / (180 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5 * Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-1) = A084448 = 1/12 - log(A074962), Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4.

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

A283264 Expansion of exp( Sum_{n>=1} -sigma_4(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -8, -19, -9, 127, 500, 1038, 448, -4967, -21463, -50043, -59084, 70418, 600080, 1837349, 3532062, 3179251, -6965009, -42260393, -119597290, -224546234, -223670132, 292245783, 2156083245, 6428174973, 13030612271, 16820582355, -133402359, -78307103593

Offset: 0

Seiichi Manyama, Mar 04 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=3 of A283272.
Cf. A023872 (exp( Sum_{n>=1} sigma_4(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), this sequence (k=4), A283271 (k=5).

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -n^3)
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{n>=1} (1 - x^n)^(n^3).

a(n) = -(1/n)*Sum_{k=1..n} sigma_4(k)*a(n-k).

A088327 G.f.: exp(Sum_{k>=1} B(x^k)/k), where B(x) = x + 2x^2 + 5x^3 + 14x^4 + 42x^5 + ... = (C(x)-1)/x and C is the g.f. for the Catalan numbers A000108.

Original entry on oeis.org

1, 1, 3, 8, 25, 77, 256, 854, 2940, 10229, 36124, 128745, 463137, 1677816, 6118165, 22432778, 82660369, 305916561, 1136621136, 4238006039, 15852603939, 59471304434, 223704813807, 843547443903, 3188064830876, 12074092672950, 45816941923597, 174173975322767

Offset: 0

N. J. A. Sloane, Nov 06 2003

nonn

a(n) is the number of forests of rooted plane binary trees (each node has outdegree = 0 or 2) where the trees have a total of n internal nodes. Cf. A222006. - Geoffrey Critzer, Feb 26 2013

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

Row sums of A275431.

Cf. A222006, A304787.

m:=35;
f:= func< x | (&*[1/(1-x^j)^Catalan(j): j in [1..m+2]]) >;
R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( f(x) )); // G. C. Greubel, Dec 12 2022

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
       binomial(2*d, d)/(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2012

With[{nn=35}, CoefficientList[Series[Product[1/(1-x^i)^CatalanNumber[i], {i,nn}], {x,0,nn}], x]] (* Geoffrey Critzer, Feb 26 2013 *)

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(2*n, n)/(n+1))
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

Euler transform of Catalan numbers (A000108). - Franklin T. Adams-Watters, Mar 01 2006

a(n) ~ c * 4^n / n^(3/2), where c = exp(Sum_{k>=1} (-2 + 4^k - 4^k*sqrt(1 - 4^(1-k)))/(2*k) ) / sqrt(Pi) = 1.60022306097485382475864802335610662545... - Vaclav Kotesovec, Mar 21 2021

A260916 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(Fibonacci(k)).

Original entry on oeis.org

1, 2, 4, 10, 22, 48, 104, 220, 460, 954, 1956, 3976, 8026, 16084, 32032, 63440, 124974, 245008, 478204, 929452, 1799508, 3471396, 6673724, 12788976, 24433528, 46546738, 88432264, 167575474, 316768948, 597389576, 1124092476, 2110661644, 3955006820, 7396477224

Offset: 0

Vaclav Kotesovec, Aug 18 2015

nonn

Convolution of A166861 and A261050.

G. C. Greubel, 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, A001622, A166861, A261050, A261519, A261520.

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

a(n) ~ phi^n / (2^(3/4) * 5^(1/8) * sqrt(Pi) * n^(3/4)) * exp(-1/5 + 2*5^(-1/4)*sqrt(2*n) + s), where s = 2 * Sum_{k>=1} phi^(2*k+1) / ((phi^(4*k+2) - phi^(2*k+1) - 1)*(2*k+1)) = 0.276751423987223411719438512082359840225908317... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A261031 Euler transform of Lucas numbers.

Original entry on oeis.org

1, 1, 4, 8, 21, 44, 103, 217, 477, 999, 2116, 4373, 9055, 18464, 37576, 75725, 152047, 303158, 602085, 1189242, 2340065, 4584027, 8947865, 17399906, 33725509, 65153150, 125493914, 241011287, 461611911, 881806114, 1680336592, 3194346093, 6058770147, 11466709780

Offset: 0

Vaclav Kotesovec, Aug 07 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A000032, A166861.

L:= proc(n) option remember; `if`(n<2, 2-n, L(n-2)+L(n-1)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      L(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 12 2017

CoefficientList[Series[Product[1/(1 - x^k)^LucasL[k], {k, 1, 30}], {x, 0, 30}], x]

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(1, 1, 2)
b = EulerTransform(a)
print([b(n) for n in range(34)]) # Peter Luschny, Nov 11 2020

a(n) ~ phi^n / (2*sqrt(Pi)*n^(3/4)) * exp(-1 + 1/(2*sqrt(5)) + 2*sqrt(n) + s), where s = Sum_{k>=2} (2 + phi^k)/((phi^(2*k) - phi^k - 1)*k) = 0.9799662013576411396292209835034813778512885279062665867878344706... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 07 2015

G.f.: exp(Sum_{k>=1} x^k*(1 + 2*x^k)/(k*(1 - x^k - x^(2*k)))). - Ilya Gutkovskiy, May 30 2018

A200544 Number of distinct bags of distinct sequences of 1s and 2s such that the sum of all terms is n.

Original entry on oeis.org

1, 1, 3, 6, 14, 28, 61, 122, 253, 505, 1017, 2008, 3976, 7769, 15169, 29379, 56751, 108993, 208725, 397913, 756385, 1432578, 2705744, 5094749, 9568504, 17922756, 33492061, 62438472, 116151352, 215612548, 399451325, 738612472, 1363261171, 2511748010, 4620024202

Offset: 0

Sarah Nibs, Nov 18 2011

nonn

This is the number of distinct ways to build minimal Jenga towers out of n blocks. The number of distinct ways to build a single minimal Jenga tower out of n blocks is the Fibonacci number F(n+1) (A000045(n+1)).
To calculate this, first create all partitions of n.
An example partition, for n=4, is
  1 1 1 1
  1 1 2
  1 3
  2 2
  4
Then each set of towers of the same size gets a configuration. For 2 2 2, for instance, there are two possibilities for each tower (a single level with two blocks or two levels with one block each) but the total possibilities is not 2*2*2=8, since the configuration "1/1,2,2" is the same as "2,1/1,2". Instead we want to choose three towers with repetition from two possibilities which is 3+2-1 choose 3, aka 4C3 = 4.
Multiply all the sets of towers of the same size and sum over partitions for the result.
For n=4, then, 1 1 2 becomes "1 with multiplicity 2" and "2 with multiplicity 1".
There is f(1+1)=1 way to build a tower of size 1, and f(1+1)+2-1 choose 2 = 2C2 = 1 way to build 2 towers of size 1. f(2+1)=2 ways to build a tower of size 2. 1 1 2 has 1*2=2 ways to be built. Sum over each of the 5 partitions of n=4.
This is apparently the limit of the row-reversed rows of the Multiset transform T(n,k) of the Fibonacci sequence in A337009, a(k) = lim_{n->oo} T(n,n-k). - R. J. Mathar, Aug 10 2020

			For n = 4, a(4)=14 and the bags are: 1/1/1/1; 1/1/1,1; 1/1/2; 1/1,1,1; 1/1,2; 1/2,1; 1,1/1,1; 1,1/2; 2/1,1; 2/2; 1,1,1,1; 1,1,2; 1,2,1; 2,1,1.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
W. S. Gray, K. Ebrahimi-Fard, Affine SISO Feedback Transformation Group and Its Faa di Bruno Hopf Algebra, arXiv:1411.0222 [math.OC], 2014.
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015
Vaclav Kotesovec, Asymptotics of sequence A034691
Sarah Nibs, C# code to generate sequence terms
Wikipedia, Jenga

Cf. A000045, A034691, A166861, A260787, A337009.

with(numtheory):with(combinat):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      fibonacci(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 05 2013

CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k+1], {k, 1, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 05 2015 *)

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(1, 1, 1)
b = EulerTransform(a)
print([b(n) for n in range(35)]) # Peter Luschny, Nov 11 2020

sum{m1*a1+m2*a2+...+mk*ak}(prod{k}(binomial(A000045[ak + 1]+mk-1,mk))).
G.f.: Product_{s>=1}(sum{d>=0}(binomial(F(s+1)+d-1,d)*x^(d*s))). - Sarah Nibs, Oct 21 2013
Euler Transform of A000045 starting at index 2, i.e. EULER(1, 2, 3, 5, 8, 13, ...). - Sarah Nibs, Nov 05 2013
a(n) ~ phi^(n+1/4) / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(phi/10 - 3/5 + 2*5^(-1/4)*sqrt(phi*n) + s), where s = Sum_{k>=2} (1+phi^k) / ((phi^(2*k) - phi^k - 1)*k) = 0.7902214013751085262994702391769374769675268259229550490716908... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 06 2015
a(n) = A337009(2*n,n). - Alois P. Heinz, Apr 30 2023

Corrected terms from n=8 and onwards by Sarah Nibs, Oct 18 2013

C# program corrected and made much more efficient by Sarah Nibs, Oct 18 2013

cp's OEIS Frontend

A030009 Euler transform of primes.

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A023880 Number of partitions in expanding space.

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A258349 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)/2).

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A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)*(k-2)/6).

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

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A283264 Expansion of exp( Sum_{n>=1} -sigma_4(n)*x^n/n ) in powers of x.

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A088327 G.f.: exp(Sum_{k>=1} B(x^k)/k), where B(x) = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ... = (C(x)-1)/x and C is the g.f. for the Catalan numbers A000108.

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Author

Keywords

Comments

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Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A260916 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(Fibonacci(k)).

Views

A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)/6).

A088327 G.f.: exp(Sum_{k>=1} B(x^k)/k), where B(x) = x + 2x^2 + 5x^3 + 14x^4 + 42x^5 + ... = (C(x)-1)/x and C is the g.f. for the Catalan numbers A000108.