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.

A047792 a(n) = Sum_{k=0..n} Stirling1(n,k)*Stirling2(n,k).

Original entry on oeis.org

1, 1, 0, -6, 36, 50, -6575, 145222, -1489978, -49083480, 4200404478, -182031111702, 4165517606173, 176264238017452, -33427749628678925, 2913726991238703330, -165770248921085801710, 1422295225609567363172, 1326793746164926878993976
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A008275, A008277, A047793, A047794, A047795.

Programs

  • GAP
    List([0..20], n-> Sum([0..n], k-> (-1)^(n-k)*Stirling1(n,k) *Stirling2(n,k) )); # G. C. Greubel, Aug 07 2019
  • Magma
    [(&+[StirlingFirst(n,k)*StirlingSecond(n,k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 07 2019
  • Maple
    seq(add(stirling1(n, k)*stirling2(n, k), k = 0..n), n = 0..20); # G. C. Greubel, Aug 07 2019
  • Mathematica
    Flatten[{1, Table[Sum[StirlingS1[n, k]*StirlingS2[n, k], {k, n}], {n,20}] }] (* Vaclav Kotesovec, Oct 13 2018 *)
  • PARI
    {a(n) = sum(k=0,n, stirling(n,k,1)*stirling(n,k,2))};
vector(20, n, n--; a(n)) \\ G. C. Greubel, Aug 07 2019
  • Sage
    [sum((-1)^(n-k)*stirling_number1(n,k)*stirling_number2(n,k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 07 2019

A047794 a(n) = Sum_{k=0..n} C(n,k)*|Stirling1(n,k)*Stirling2(n,k)|.

Original entry on oeis.org

1, 1, 3, 34, 631, 16871, 617356, 28968990, 1680536159, 117572734195, 9715771690081, 932711356031016, 102653506699902874, 12810868034079756421, 1795954763065584594656, 280569433733767673934426, 48506369621902094002862671, 9224242346164172284054561019
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A008275, A008277, A047792, A047793, A047795.

Programs

  • GAP
    List([0..20], n-> Sum([0..n], k-> Stirling1(n,k)*Stirling2(n,k) *Binomial(n,k) )); # G. C. Greubel, Aug 07 2019
  • Magma
    [(&+[(-1)^(n-k)*StirlingFirst(n,k)*StirlingSecond(n,k) *Binomial(n,k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 07 2019
  • Maple
    seq(add((-1)^(n-k)*binomial(n, k)*stirling1(n, k)*stirling2(n, k), k = 0 .. n), n = 0..20); # G. C. Greubel, Aug 07 2019
  • Mathematica
    Table[Sum[Binomial[n,k]Abs[StirlingS1[n,k]StirlingS2[n,k]],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Apr 10 2012 *)
  • PARI
    {a(n) = sum(k=0,n, (-1)^(n-k)*stirling(n,k,1)*stirling(n,k,2) *binomial(n,k))};
vector(20, n, n--; a(n)) \\ G. C. Greubel, Aug 07 2019
  • Sage
    [sum(stirling_number1(n,k)*stirling_number2(n,k)*binomial(n,k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 07 2019

A047795 a(n) = Sum_{k=0..n} C(n,k)*Stirling1(n,k)*Stirling2(n,k).

Original entry on oeis.org

1, 1, -1, -20, 295, 871, -196784, 6287772, 29169631, -18200393741, 1304183716981, -27109895360074, -6212943553813622, 1062831339757496245, -85292203894284124100, -1487854700305245210924, 1896933688279584387159631, -377233175400513002923379973
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A008275, A008277, A047792, A047793, A047794.

Programs

  • GAP
    List([0..20], n-> Sum([0..n], k-> (-1)^(n-k)*Stirling1(n,k) *Stirling2(n,k)*Binomial(n,k) )); # G. C. Greubel, Aug 07 2019
  • Magma
    [(&+[StirlingFirst(n,k)*StirlingSecond(n,k)*Binomial(n,k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 07 2019
  • Maple
    seq(add(binomial(n,k)*stirling1(n,k)*stirling2(n, k), k = 0..n), n = 0 .. 20); # G. C. Greubel, Aug 07 2019
  • Mathematica
    Table[Sum[Binomial[n, k]*StirlingS1[n, k]*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] (* G. C. Greubel, Aug 07 2019 *)
  • PARI
    {a(n) = sum(k=0,n,stirling(n,k,1)*stirling(n,k,2)*binomial(n,k))};
vector(20, n, n--; a(n)) \\ G. C. Greubel, Aug 07 2019
  • Sage
    [sum((-1)^(n-k)*stirling_number1(n,k)* stirling_number2(n,k) *binomial(n,k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 07 2019

A192546 a(n) = sum(stirling2(n,k)*stirling2(n+1,k+1),k=0..n).

Original entry on oeis.org

1, 1, 4, 26, 251, 3157, 50310, 978315, 22616102, 610543614, 18953178234, 668200802513, 26484030901784, 1169889631517219, 57168091338306720, 3070545537985858612, 180251920018830890897, 11507450028966272232867, 795397552661209049095698
Offset: 0

Views

Author

Emanuele Munarini, Jul 04 2011

Keywords

Links

Crossrefs

Cf. A047793

Programs

  • Mathematica
    Table[Sum[StirlingS2[n,k]StirlingS2[n+1,k+1],{k,0,n}],{n,0,100}]
Total/@Table[StirlingS2[n,k]StirlingS2[n+1,k+1],{n,0,20},{k,0,n}] (* Harvey P. Dale, Dec 15 2023 *)
  • Maxima
    makelist(sum(stirling2(n,k)*stirling2(n+1,k+1),k,0,n),n,0,24);

A192548 a(n) = sum(abs(stirling1(n,k))*stirling2(n+1,k+1),k=0..n).

Original entry on oeis.org

1, 1, 4, 33, 426, 7670, 181000, 5376777, 195238792, 8472419484, 431606519268, 25440239275308, 1714357181128372, 130748750027622922, 11188498960336877296, 1066226987215138587095, 112415085220156146401380, 13037223283297354475403696
Offset: 0

Views

Author

Emanuele Munarini, Jul 04 2011

Keywords

Links

Crossrefs

Cf. A047793.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n,k]]StirlingS2[n+1,k+1],{k,0,n}],{n,0,100}]
  • Maxima
    makelist(sum(abs(stirling1(n,k))*stirling2(n+1,k+1),k,0,n),n,0,24);

A192564 a(n) = Sum_{k=0..n} abs(Stirling1(n,k))*Stirling2(n,k)*(k!)^2.

Original entry on oeis.org

1, 1, 5, 74, 2186, 106524, 7703896, 773034912, 102673179360, 17429291711280, 3680338415133024, 945958227345434016, 290761516548473591232, 105309706114422166775040, 44384982810939832477305600, 21536846291826596564956445184
Offset: 0

Views

Author

Emanuele Munarini, Jul 04 2011

Keywords

Crossrefs

Cf. A047793, A048144, A382792.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n,k]]StirlingS2[n,k]k!^2,{k,0,n}],{n,0,100}]
nmax = 20; Table[SeriesCoefficient[1/(1 + (E^x - 1)*Log[1 - y]), {x, 0, n}, {y, 0, n}], {n, 0, nmax}] * Range[0, nmax]!^2 (* Vaclav Kotesovec, Apr 08 2025 *)
  • Maxima
    makelist(sum(abs(stirling1(n,k))*stirling2(n,k)*k!^2,k,0,n),n,0,24);

Formula

a(n) ~ c * LambertW(-1, -r*exp(-r))^n * n!^2 / (sqrt(n) * LambertW(-exp(-1/r)/r)^n), where r = 0.673313285145753168... is the root of the equation (1 + 1/(r*LambertW(-exp(-1/r)/r))) * (r + LambertW(-1, -r*exp(-r))) = 1 and c = 0.27034346270211507329954765593360596752557904498770241464597402478625037569... - Vaclav Kotesovec, Jul 05 2021
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 + (exp(x) - 1) * log(1 - y)). - Ilya Gutkovskiy, Apr 06 2025

A384501 a(n) = Sum_{k=0..n} abs(Stirling1(n,k)) * Stirling2(n,n-k).

Original entry on oeis.org

1, 0, 1, 9, 119, 2025, 42510, 1062761, 30854159, 1020615912, 37900765365, 1561459425955, 70682817696436, 3487456195458027, 186281997929231659, 10709829446929099865, 659427284782849503663, 43293574636994934145044, 3019108475859713906967738, 222868205832269470083471366
Offset: 0

Views

Author

Vaclav Kotesovec, May 31 2025

Keywords

Crossrefs

Cf. A047793, A187655, A187656.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n, k]]*StirlingS2[n, n-k], {k, 0, n}], {n, 0, 20}]
Table[Sum[StirlingS2[n, k]*Abs[StirlingS1[n, n-k]], {k, 0, n}], {n, 0, 20}]

Formula

a(n) = Sum_{k=0..n} abs(Stirling1(n,n-k)) * Stirling2(n,k).
a(n) ~ c * ((-r - 1/((1-r)*LambertW(exp(1/(r-1))/(r-1)))) / (1 + (1-r)*LambertW(exp(1/(r-1))/(r-1))))^n * n^(n - 1/2) / exp(n), where r = 0.412059483521755003540032983286575579547027818844750... is the root of the equation (1-r)^2 * (1 + LambertW(-1, -exp(-r)*r)/r) = (1-r) + 1/LambertW(exp(1/(r-1))/(r-1)) and c = 0.21367572159147979376975234273...
Showing 1-7 of 7 results.

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

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