cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A058694 Partial products p(0)*p(1)*...*p(n) of partition numbers A000041.

Original entry on oeis.org

1, 1, 2, 6, 30, 210, 2310, 34650, 762300, 22869000, 960498000, 53787888000, 4141667376000, 418308404976000, 56471634671760000, 9939007702229760000, 2295910779215074560000, 681885501426877144320000, 262525918049347700563200000, 128637699844180373275968000000
Offset: 0

Views

Author

N. J. A. Sloane, Dec 30 2000

Keywords

Comments

a(n) gives the number of partitions P(V(n)) of V(n)=[1,2,3,...,n]. A partition P(V(n)) acts on the components of V(n), i.e., the components of V(n) are partitioned. Therefore a(n) results as the product of the number of partitions P(i) of the component v(i)=i with i=1,...,n. For example, a(3) = 6 because we have 6 list partitions for the list V(n=3)=[1,2,3]: [[1], [1, 1], [2, 1]], [[1], [1, 1], [1, 1, 1]], [[1], [1, 1], [3]], [[1], [2], [2, 1]], [[1], [2], [1, 1, 1]], [[1], [2], [3]]. - Thomas Wieder, Sep 29 2007
Equals the eigensequence of triangle A174712; i.e., Triangle A174712 * A058694 preceded by a 1 shifts left. - Gary W. Adamson, Mar 27 2010

Links

Crossrefs

Cf. A000041, A000070, A133018, A259314, A259373, A174712.

Programs

  • Maple
    a:= proc(n) option remember;
       combinat[numbpart](n)*`if`(n>0, a(n-1), 1)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 21 2012
#
# The constant S in the Maple notation
evalf(Zeta(0, -1/2, 23/24)*sqrt(2/3)*Pi - Zeta(0, 1/2, 23/24)*sqrt(3/2)/Pi+3*(D(GAMMA))(23/24)/(4*Pi^2*GAMMA(23/24)) - (Sum(Zeta(0, j/2, 23/24)*(sqrt(3/2)/Pi)^j/j, j=3..infinity)), 60); # Vaclav Kotesovec, Jun 24 2015
  • Mathematica
    Table[Product[PartitionsP[k], {k, 1, n}], {n, 1, 33}] (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
  • PARI
    a(n)=prod(k=2,n, numbpart(k)) \\ Charles R Greathouse IV, Jan 14 2017

Formula

a(n) ~ C * Product_{k=1..n} (exp(Pi*sqrt(2/3*(k-1/24))) / (4*sqrt(3)*(k-1/24)) * (1 - sqrt(3/(2*(k-1/24)))/Pi)), where C = 0.9110167313322499518... is the partition factorial constant A259314. - Vaclav Kotesovec, Jun 24 2015
a(n) ~ C * Gamma(23/24) / (n^(n + 11/24 + 3/(4*Pi^2)) * 2^(2*n) * 3^(n/2) * sqrt(2*Pi)) * exp(Pi*(2*n/3)^(3/2) + n + (11*Pi/(12*sqrt(6)) - sqrt(6)/Pi)*sqrt(n) + S), where C = A259314 and S = Zeta(-1/2, 23/24)*sqrt(2/3)*Pi - Zeta(1/2, 23/24)*sqrt(3/2)/Pi + 3*Gamma'(23/24)/(4*Pi^2*Gamma(23/24)) - Sum_{j>=3} Zeta(j/2, 23/24)*(sqrt(3/2)/Pi)^j/j = -0.02541933397793652709903012019225640813047573968579474..., Zeta is the Hurwitz Zeta Function, in Maple notation Zeta(0,z,v), in Mathematica notation Zeta[z,v], equivalently HurwitzZeta[z,v]. - Vaclav Kotesovec, Jun 24 2015

A259373 a(n) = Product_{k=0..n} p(k)^k, where p(k) is the partition function A000041.

Original entry on oeis.org

1, 1, 4, 108, 67500, 1134472500, 2009787236572500, 343390991123754492187500, 18843880602308850038793150000000000, 370904101895245095313565571450000000000000000000, 6335115544513765517772271190776403515352524800000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 25 2015

Keywords

Crossrefs

Cf. A000041, A058694, A133018, A259314, A259405, A265097.

Programs

  • Mathematica
    Table[Product[PartitionsP[k]^k,{k,0,n}],{n,0,10}]

Formula

a(n) ~ c * Product_{k=1..n} ( (exp(Pi*sqrt(2/3*(k-1/24))) / (4*sqrt(3)*(k-1/24)) * (1 - sqrt(3/(2*(k-1/24)))/Pi)) )^k, where c = A259405 = 0.90866166764445489256...

A265094 a(n) = q(n)^n, where q(n) = partition numbers into distinct parts (A000009).

Original entry on oeis.org

1, 1, 1, 8, 16, 243, 4096, 78125, 1679616, 134217728, 10000000000, 743008370688, 129746337890625, 20822964865671168, 6221821273427820544, 2954312706550833698643, 1208925819614629174706176, 718325266223569592115396608, 850434696123579966501779931136
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 01 2015

Keywords

Links

Crossrefs

Cf. A133018.

Programs

  • Mathematica
    Table[PartitionsQ[n]^n,{n,0,20}]

Formula

a(n) ~ exp(n^(3/2)*Pi/sqrt(3) + (Pi/(48*sqrt(3)) - 3*sqrt(3)/(8*Pi))*sqrt(n) - 1/32 - 9/(16*Pi^2)) / (3^(n/4) * 4^n * n^(3*n/4)).

A197987 a(n) = prime(n)^(n+1).

Original entry on oeis.org

4, 27, 625, 16807, 1771561, 62748517, 6975757441, 322687697779, 41426511213649, 12200509765705829, 787662783788549761, 243569224216081305397, 37929227194915558802161, 3177070365797955661914307, 566977372488557307219621121, 205442259656281392806087233013
Offset: 1

Views

Author

Bruno Berselli, Oct 20 2011

Keywords

Comments

Subsequence of A000961, A120458.
First five elements are also consecutive members of A133018. - Omar E. Pol, Oct 20 2011
Third diagonal of A319075. - Omar E. Pol, Sep 13 2018

Examples

			The fourth prime number is 7, so a(4) = 7^(4+1) = 7^5 = 16807. - _Omar E. Pol_, Oct 20 2011

Links

Crossrefs

Cf. A000961, A062457, A093360, A120458, A133018, A319075.

Programs

  • Magma
    [NthPrime(n)^(n+1): n in [1..16]];
  • Mathematica
    Table[Prime[n]^(n+1),{n,20}] (* Harvey P. Dale, Dec 16 2012 *)
  • PARI
    for(n=1, 16, print1(prime(n)^(n+1)", "));

Formula

a(n) = A000040(n)^(n+1). - Omar E. Pol, Oct 20 2011

A259405 Decimal expansion of a constant related to A259373.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Jun 26 2015

Keywords

Examples

			0.908661667644454892566581137702159278136942213727370666511234283397226865...

Crossrefs

Cf. A000041, A259373, A058694, A259314, A133018.

Programs

  • Mathematica
    (* The iteration cycle: *) Do[Print[Product[N[PartitionsP[k]^k/((E^(Sqrt[2/3]*Sqrt[k-1/24]*Pi) * (1 - Sqrt[3/2]/(Sqrt[k-1/24]*Pi))) / (4*Sqrt[3]*(k-1/24)))^k, 150], {k, 1, n}]], {n, 1000, 50000, 1000}]

Formula

Equals limit n->infinity Product_{k=1..n} p(k)^k / (exp(Pi*sqrt(2/3*(k-1/24))) / (4*sqrt(3)*(k-1/24)) * (1 - sqrt(3/(2*(k-1/24)))/Pi))^k, where p(k) is the partition function A000041.

A259436 a(n) = Sum_{k=0..n} p(k)^k, where p(k) is the partition function A000041.

Original entry on oeis.org

1, 2, 6, 33, 658, 17465, 1789026, 172648401, 55048521937, 19738048521937, 17099936170199761, 17002207325552593617, 43456890729289136241538, 113852784934058230923022839, 667954362620824922543667163464, 4816707198961510396593071163584840
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 27 2015

Keywords

Crossrefs

Cf. A000041, A133018, A259373, A259399, A259437, A259438, A265095.

Programs

  • Mathematica
    Table[Sum[PartitionsP[k]^k,{k,0,n}],{n,0,15}]

Formula

a(n) ~ p(n)^n ~ exp(1/24 - 3/(4*Pi^2) - (72+Pi^2)*sqrt(n)/(24*sqrt(6)*Pi) + sqrt(2/3)*Pi*n^(3/2)) / (3^(n/2) * 4^n * n^n).

A259437 a(n) = Sum_{k=0..n} p(k)^n, where p(k) is the partition function A000041.

Original entry on oeis.org

1, 2, 6, 37, 724, 20209, 1905630, 191250531, 57659285287, 20931112851787, 17697850924585423, 17720783665888137843, 44421728434157120665320, 117208746422032553556330253, 679595843556865572365153402674, 4907378683411420479410336076467628
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 27 2015

Keywords

Crossrefs

Cf. A000041, A133018, A259399, A259436, A259438.

Programs

  • Mathematica
    Table[Sum[PartitionsP[k]^n,{k,0,n}],{n,0,15}]

Formula

a(n) ~ p(n)^n ~ exp(1/24 - 3/(4*Pi^2) - (72+Pi^2)*sqrt(n)/(24*sqrt(6)*Pi) + sqrt(2/3)*Pi*n^(3/2)) / (3^(n/2) * 4^n * n^n).

A259438 a(n) = Sum_{k=0..n} p(k)^(n-k), where p(k) is the partition function A000041.

Original entry on oeis.org

1, 2, 3, 5, 10, 25, 78, 301, 1414, 7964, 53408, 426116, 4028890, 44697755, 576491980, 8617031811, 149425700853, 3004591733938, 69763130950599, 1860330686377532, 56746090401472922, 1975156902590115291, 78299783319570477185, 3529323014512112469681
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 27 2015

Keywords

Comments

The position of the maximum value asymptotically approaches k = n/3.

Crossrefs

Cf. A000041, A133018, A259399, A259436, A259437.

Programs

  • Mathematica
    Table[Sum[PartitionsP[k]^(n-k),{k,0,n}],{n,0,25}]

Formula

log(a(n)) ~ 2^(3/2)*Pi*n^(3/2)/9 - n*log(16*n^2/3)/3.
G.f.: Sum_{k>=0} x^k/(1 - p(k)*x). - Ilya Gutkovskiy, Oct 09 2018

A265015 a(n) = A015128(n)^n.

Original entry on oeis.org

1, 2, 16, 512, 38416, 7962624, 4096000000, 4398046511104, 10000000000000000, 48717667557975775744, 451730952053751361306624, 7982572438812891719395180544, 268637376395543538746286686601216, 16132732437821617561429013924830773248
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 05 2015

Keywords

Links

Crossrefs

Cf. A015128, A133018, A156616, A265094.

Programs

  • Mathematica
    nmax = 15; A015128 = Rest[CoefficientList[Series[Product[(1+x^k)/(1-x^k), {k,1,nmax}], {x,0,nmax}], x]]; Flatten[{1, Table[A015128[[n]]^n, {n,1,nmax}]}]

Formula

a(n) ~ exp(Pi*n^(3/2) - sqrt(n)/Pi - 1/(2*Pi^2)) / (8^n * n^n) * (1 - 1/(3*Pi^3*sqrt(n))).
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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