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 1-6 of 6 results.

A007113 Expansion of e.g.f. (1 + x)^x.

Original entry on oeis.org

1, 0, 2, -3, 20, -90, 594, -4200, 34544, -316008, 3207240, -35699400, 432690312, -5672581200, 79991160144, -1207367605080, 19423062612480, -331770360922560, 5997105160795584, -114373526841360000, 2295170834453089920, -48344592370577247360
Offset: 0

Views

Author

Simon Plouffe

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.3.

Links

Crossrefs

Cf. A053489, A053490. Apart from initial terms and signs, same as A066166.

Programs

  • Maple
    a:= n-> n! *coeff(series((1+x)^x, x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 12 2012
  • Mathematica
    CoefficientList[Series[(1 + x)^x, {x, 0, 19}], x]*Table[(n - 1)!, {n, 1, 20}]
a[n_] := (-1)^n*n!*Sum[ StirlingS1[n - k, k]/(n - k)!*(-1)^(n - 2*k), {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Dec 12 2012, after Vladeta Jovovic *)

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*k!*Stirling1(n-k, k). - Vladeta Jovovic, Dec 19 2004
a(n) ~ (-1)^n * n!. - Vaclav Kotesovec, Jun 06 2019

Extensions

Signs from Christian G. Bower, Nov 15 1998

A053489 Expansion of e.g.f.: (1-x)^(-2x).

Original entry on oeis.org

1, 0, 4, 6, 64, 300, 2568, 20160, 193856, 1989792, 22687200, 279956160, 3737966208, 53589444480, 821522026752, 13407498599040, 232106716968960, 4248256958023680, 81968803604600832, 1662870215019018240, 35384007384670648320, 788053048823608565760
Offset: 0

Views

Author

N. J. A. Sloane, Jan 15 2000

Keywords

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.3.

Links

Crossrefs

Cf. A007113, A053490.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-x)^(-2*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 29 2018
  • Mathematica
    CoefficientList[Series[(1-x)^(-2*x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Apr 21 2014 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace((1-x)^(-2*x))) \\ G. C. Greubel, Aug 29 2018

Formula

E.g.f.: (1-x)^(-2*x).
a(n) = (-1)^n*Sum_{k=0..floor(n/2)} 2^k*binomial(n, k)*k!*Stirling1(n-k, k). - Vladeta Jovovic, Dec 19 2004
a(n) ~ n! * n * (1 + (1-2*log(n)-2*gamma)/n), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Apr 21 2014

A318615 a(n) = n! * [x^n] 1/(1 - x)^(n*x).

Original entry on oeis.org

1, 0, 4, 9, 224, 1650, 38664, 540960, 13930496, 291769128, 8598924000, 237964577400, 8082061452288, 275311724996880, 10714824398213376, 430458433091505000, 19007133744632954880, 876046954673290438080, 43416883192646088235008, 2252711496770428822876800, 124040138653975179571200000
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 30 2018

Keywords

Links

Crossrefs

Cf. A007113, A008275, A053489, A053490, A191415, A318616.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(1 - x)^(n x), {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[(-1)^n n! Sum[n^(n - k) StirlingS1[k, n - k]/k!, {k, n}], {n, 20}]]

Formula

a(n) = n! * [x^n] exp(n*x*Sum_{k>=1} x^k/k).
a(n) = (-1)^n*n! * Sum_{k=0..n} n^(n-k)*Stirling1(k,n-k)/k!.
a(n) ~ n^n / (sqrt(1 - (1-s)*(2-s)*s) * exp(n) * s^n * (1-s)^(s*n - 1)), where s = 0.530402312512063468084914246777198746... is the root of the equation (1-s)*(2 + s + s*log(1-s)) = 1. - Vaclav Kotesovec, Aug 30 2018

A351734 Expansion of e.g.f. exp( 3 * x * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, 6, 9, 120, 555, 5148, 39711, 378528, 3715011, 39838260, 452684463, 5463506304, 69553644771, 930940368036, 13054086036855, 191222363275968, 2918620069099395, 46309955947643124, 762335523354333855, 12995722456718984160, 229045407317491457763
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A052506, A351733.
Cf. A053490, A351737.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[3x (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 02 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(3*x*(exp(x)-1))))
  • PARI
    a(n) = n!*sum(k=0, n\2, 3^k*stirling(n-k, k, 2)/(n-k)!);

Formula

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

A351735 Expansion of e.g.f. 1/(1 - 3 * x)^x.

Original entry on oeis.org

1, 0, 6, 27, 324, 4050, 64962, 1224720, 26776656, 665390376, 18529576200, 571602980520, 19349098690248, 713092338026640, 28422424039182768, 1218246132898693080, 55876497505161154560, 2730710993688989519040, 141654212475516155694528
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A053490, A053491.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x)^x))
  • PARI
    a(n) = n!*sum(k=0, n\2, 3^(n-k)*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-k) * |Stirling1(n-k,k)|/(n-k)!.
a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 1/6) / (Gamma(1/3) * exp(n)). - Vaclav Kotesovec, Aug 30 2025

A362834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * n! * Sum_{j=0..floor(n/2)} k^j * Stirling1(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 6, 20, 0, 1, 0, 8, 9, 64, 90, 0, 1, 0, 10, 12, 132, 300, 594, 0, 1, 0, 12, 15, 224, 630, 2568, 4200, 0, 1, 0, 14, 18, 340, 1080, 6642, 20160, 34544, 0, 1, 0, 16, 21, 480, 1650, 13536, 55440, 193856, 316008, 0
Offset: 0

Views

Author

Seiichi Manyama, May 05 2023

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,    1,    1, ...
  0,  0,   0,   0,    0,    0, ...
  0,  2,   4,   6,    8,   10, ...
  0,  3,   6,   9,   12,   15, ...
  0, 20,  64, 132,  224,  340, ...
  0, 90, 300, 630, 1080, 1650, ...

Crossrefs

Columns k=0..3 give: A000007, A066166, A053489, A053490.
Main diagonal gives A318615.
Cf. A361652.

Programs

  • PARI
    T(n, k) = (-1)^n*n!*sum(j=0, n\2, k^j*stirling(n-j, j, 1)/(n-j)!);

Formula

E.g.f. of column k: 1/(1 - x)^(k*x).
Showing 1-6 of 6 results.

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

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