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 1-7 of 7 results.

A226775 Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.

Original entry on oeis.org

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

Views

Author

José María Grau Ribas, Jun 17 2013

Keywords

Examples

			-0.4063757399599599076769581241248397582109975751811406350004954883....

Links

Crossrefs

Cf. A106533, A187655, A187657, A217899, A217900, A226750, A243227, A245109, A247238, A258467, A337458.

Programs

  • Mathematica
    RealDigits[N[ProductLog[-2/E^2], 105]][[1]] (* corrected by Vaclav Kotesovec, Feb 21 2014 *)
  • PARI
    solve(x=-1, x=0, x*exp(x) + 2*exp(-2)) \\ G. C. Greubel, Nov 15 2017

Formula

Equals -2*A106533.
Equals LambertW(-2*exp(-2)).

A384471 a(n) = Sum_{k=0..n} binomial(n,k)^2 * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k).

Original entry on oeis.org

1, 2, 18, 306, 8046, 296100, 14307254, 865996306, 63308257198, 5432272670376, 535074966419260, 59461066810476232, 7354069129792197762, 1001371912804041913056, 148806933109572134044158, 23958722845801073318076450, 4154065510530807075869275150, 771608888261061026185781127184
Offset: 0

Views

Author

Vaclav Kotesovec, May 30 2025

Keywords

Crossrefs

Cf. A187655, A187657, A384470, A384472.
Cf. A226775.

Programs

  • Mathematica
    Table[Sum[StirlingS2[2*k, k]*StirlingS2[2*n-2*k, n-k]*Binomial[n, k]^2, {k, 0, n}], {n, 0, 20}]

Formula

a(n) ~ 2^(3*n + 1/2) * n^(n - 3/2) / (Pi^(3/2) * (1-w) * exp(n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...

A384472 a(n) = Sum_{k=0..n} binomial(n,k)^3 * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k).

Original entry on oeis.org

1, 2, 22, 558, 25506, 1770300, 166190354, 19647687682, 2798281247682, 466166725448544, 88942246964278060, 19127775950813311232, 4578817457796314714502, 1207681779462031251096888, 348018457509475159702959174, 108798555057988053563408904750, 36676526343321856806298038370210
Offset: 0

Views

Author

Vaclav Kotesovec, May 30 2025

Keywords

Comments

In general, for m > 1, Sum_{k=0..n} binomial(n,k)^m * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) ~ 2^((m+1)*n + (m-1)/2) * n^(n-(m+1)/2) / (sqrt(m-1) * Pi^((m+1)/2) * (1-w) * exp(n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775.

Crossrefs

Cf. A187655 (m=0), A187657 (m=1), A384471 (m=2), A384470.
Cf. A226775.

Programs

  • Mathematica
    Table[Sum[StirlingS2[2*k, k]*StirlingS2[2*n-2*k, n-k]*Binomial[n, k]^3, {k, 0, n}], {n, 0, 20}]

Formula

a(n) ~ 2^(4*n + 1/2) * n^(n-2) / (Pi^2 * (1-w) * exp(n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...

A258467 Number of partitions of 2n into parts of exactly n sorts which are introduced in ascending order.

Original entry on oeis.org

1, 2, 12, 130, 2216, 52078, 1558219, 56524414, 2406802476, 117575627562, 6478447651345, 397345158550386, 26842747368209994, 1980156804133210116, 158365138356099680582, 13647670818304698139989, 1260732993182758276252088, 124273946254095006307105363
Offset: 0

Views

Author

Alois P. Heinz, May 30 2015

Keywords

Links

Crossrefs

Cf. A256130, A187655, A187657, A217899, A217900, A245109, A319732.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(2*n, n):
seq(a(n), n=0..20);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; a[n_] := T[2n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 06 2017, translated from Maple *)

Formula

a(n) = A256130(2n,n).
a(n) ~ 2^(2*n-1/2) * n^(n-1/2) / (sqrt(Pi*(1-c)) * exp(n) * c^n * (2-c)^n), where c = -A226775 = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 31 2015
a(n) ~ Stirling2(2*n, n) = A007820(n). - Vaclav Kotesovec, Jun 01 2015

A384470 a(n) = n! * Sum_{k=0..n} Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) / binomial(n,k).

Original entry on oeis.org

1, 2, 29, 1108, 82924, 10302768, 1917699552, 499332175200, 173242955039616, 77238974345915520, 43027312823342164800, 29285800226400628915200, 23913110797474508388449280, 23071378298963178620672409600, 25964692904608781751347296204800, 33711625062334209438536728660070400
Offset: 0

Views

Author

Vaclav Kotesovec, May 30 2025

Keywords

Crossrefs

Cf. A187655, A187657, A384471, A384472.
Cf. A226775.

Programs

  • Mathematica
    Table[n! * Sum[StirlingS2[2*k, k] * StirlingS2[2*n-2*k, n-k] / Binomial[n, k], {k, 0, n}], {n, 0, 20}]

Formula

a(n) ~ 2^(2*n+1) * n^(2*n) / (sqrt(1-w) * exp(2*n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...

A384491 a(n) = n!^2 * Sum_{k=0..n} Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) / binomial(n,k)^2.

Original entry on oeis.org

1, 2, 57, 6536, 1966816, 1226860992, 1373652478656, 2507498281198080, 6966291361870181376, 27969794062091821670400, 155875927262331497576140800, 1167389777699203314381963264000, 11441270265465265986005655905894400, 143525982910350708912088976768630784000
Offset: 0

Views

Author

Vaclav Kotesovec, May 31 2025

Keywords

Crossrefs

Cf. A187655, A384470, A384492.
Cf. A187657, A384471, A384472.

Programs

  • Mathematica
    Table[n!^2 * Sum[StirlingS2[2*k, k] * StirlingS2[2*n-2*k, n-k] / Binomial[n, k]^2, {k, 0, n}], {n, 0, 15}]
  • PARI
    a(n) = n!^2 * sum(k=0, n, stirling(2*k,k, 2) * stirling(2*n-2*k,n-k,2) / binomial(n,k)^2); \\ Michel Marcus, May 31 2025

Formula

a(n) ~ sqrt(Pi) * 2^(2*n + 3/2) * n^(3*n + 1/2) / (sqrt(1-w) * exp(3*n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...

A384492 a(n) = n!^3 * Sum_{k=0..n} Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) / binomial(n,k)^3.

Original entry on oeis.org

1, 2, 113, 38992, 47071264, 147015606528, 988250901343488, 12631667044878213120, 280790763724247161061376, 10147405862241529912885248000, 565550513462476798468573003776000, 46592777163703224212146175606784000000, 5479872142880875751798643810680954683392000
Offset: 0

Views

Author

Vaclav Kotesovec, May 31 2025

Keywords

Crossrefs

Cf. A187655, A384470, A384491.
Cf. A187657, A384471, A384472.

Programs

  • Mathematica
    Table[n!^3 * Sum[StirlingS2[2*k, k] * StirlingS2[2*n-2*k, n-k] / Binomial[n, k]^3, {k, 0, n}], {n, 0, 15}]
  • PARI
    a(n) = n!^3 * sum(k=0, n, stirling(2*k,k,2) * stirling(2*n-2*k,n-k,2) / binomial(n,k)^3); \\ Michel Marcus, May 31 2025

Formula

a(n) ~ Pi * 2^(2*n+2) * n^(4*n+1) / (sqrt(1-w) * exp(4*n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...
Showing 1-7 of 7 results.

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