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 1-5 of 5 results.

A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 390, 2562, 11816, 105912, 1063530, 8815070, 81342492, 895185876, 9971185406, 112642410090, 1372455608400, 17750397057392, 236950003516626, 3286258330135734, 47688868443593540, 719345273005797900, 11222288509573985382, 181168865439054099266
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 06 2014

Keywords

Links

Crossrefs

Column k=2 of A292892.
Cf. A000110, A052506, A216507, A060311.
Cf. A216688, A216689, A080108, A000248.

Programs

  • Mathematica
    CoefficientList[Series[E^(x^2*(E^x-1)), {x, 0, 20}], x] * Range[0, 20]!
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(x^2*(exp(x) - 1)))) \\ G. C. Greubel, Nov 21 2017
  • PARI
    a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(n-2*k)!); \\ Seiichi Manyama, Jul 09 2022
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

Formula

a(n) ~ exp((n-r^3)/(2+r)-n) * n^(n+1/2) / (r^n * sqrt((2*r^3*(3+r) + n*(1+r)*(4+r))/(2+r))), where r is the root of the equation r^2*((2+r) * exp(r) - 2) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(n-2*k)!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!. (End)

A355607 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, -3, 0, 1, 0, 0, 6, 20, 0, 1, 0, 0, 0, -12, -90, 0, 1, 0, 0, 0, 24, 40, 594, 0, 1, 0, 0, 0, 0, -60, 180, -4200, 0, 1, 0, 0, 0, 0, 120, 240, -1512, 34544, 0, 1, 0, 0, 0, 0, 0, -360, -1260, 11760, -316008, 0, 1, 0, 0, 0, 0, 0, 720, 1680, 28224, -38880, 3207240, 0
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2022

Keywords

Examples

			Square array begins:
  1,   1,   1,   1,    1,   1, 1, ...
  1,   0,   0,   0,    0,   0, 0, ...
  0,   2,   0,   0,    0,   0, 0, ...
  0,  -3,   6,   0,    0,   0, 0, ...
  0,  20, -12,  24,    0,   0, 0, ...
  0, -90,  40, -60,  120,   0, 0, ...
  0, 594, 180, 240, -360, 720, 0, ...

Links

Crossrefs

Columns k=1..4 give A007113, A007121, (-1)^n * A353229(n), A354625.
Cf. A292892, A355609, A355619.

Programs

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

Formula

T(0,k) = 1 and T(n,k) = -(n-1)! * Sum_{j=k+1..n} (-1)^(j-k) * j/(j-k) * T(n-j,k)/(n-j)! for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling1(n-k*j,j)/(n-k*j)!.

A292891 Expansion of e.g.f. exp(x^3 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 120, 210, 20496, 181944, 1059120, 4990590, 100458600, 1634594676, 18436740504, 164378216730, 2124284725920, 38171412643440, 631390188466656, 8760417873485814, 124649582165430840, 2167585391936047020, 41833303600025220360
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2017

Keywords

Links

Crossrefs

Column k=3 of A292892.
Cf. A292889, A354001.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x^3 (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 21 2022 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(x^3*(exp(x)-1))))
  • PARI
    a(n) = n!*sum(k=0, n\4, stirling(n-3*k, k, 2)/(n-3*k)!); \\ Seiichi Manyama, Jul 09 2022
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=4, i, j/(j-3)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

Formula

From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/4)} Stirling2(n-3*k,k)/(n-3*k)!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=4..n} k/(k-3)! * a(n-k)/(n-k)!. (End)

A292894 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (1 - exp(x))).

Original entry on oeis.org

1, 1, -1, 1, 0, 0, 1, 0, -2, 1, 1, 0, 0, -3, 1, 1, 0, 0, -6, 8, -2, 1, 0, 0, 0, -12, 55, -9, 1, 0, 0, 0, -24, -20, 84, -9, 1, 0, 0, 0, 0, -60, 330, -637, 50, 1, 0, 0, 0, 0, -120, -120, 2478, -4992, 267, 1, 0, 0, 0, 0, 0, -360, -210, 11704, -10593, 413, 1, 0, 0, 0, 0, 0, -720, -840, 19824, -15192, 92060, -2180
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2017

Keywords

Examples

			Square array begins:
   1,  1,   1,   1,    1, ...
  -1,  0,   0,   0,    0, ...
   0, -2,   0,   0,    0, ...
   1, -3,  -6,   0,    0, ...
   1,  8, -12, -24,    0, ...
  -2, 55, -20, -60, -120, ...

Links

Crossrefs

Columns k=0..2 give A000587, A292893, A292951.
Rows n=0..1 give A000012, (-1)*A000007.
Main diagonal gives A000007.
Cf. A292892, A355607.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), (-1)^j*stirling(n-k*j, j, 2)/(n-k*j)!); \\ Seiichi Manyama, Jul 09 2022

Formula

From Seiichi Manyama, Jul 09 2022: (Start)
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} (-1)^j * Stirling2(n-k*j,j)/(n-k*j)!.
T(0,k) = 1 and T(n,k) = -(n-1)! * Sum_{j=k+1..n} j/(j-k)! * T(n-j,k)/(n-j)! for n > 0. (End)

A355650 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k/k! * (exp(x) - 1)).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 2, 5, 1, 0, 0, 3, 15, 1, 0, 0, 3, 16, 52, 1, 0, 0, 0, 6, 65, 203, 1, 0, 0, 0, 4, 10, 336, 877, 1, 0, 0, 0, 0, 10, 105, 1897, 4140, 1, 0, 0, 0, 0, 5, 20, 651, 11824, 21147, 1, 0, 0, 0, 0, 0, 15, 35, 2968, 80145, 115975, 1, 0, 0, 0, 0, 0, 6, 35, 616, 18936, 586000, 678570
Offset: 0

Views

Author

Seiichi Manyama, Jul 12 2022

Keywords

Examples

			Square array begins:
    1,   1,   1,  1,  1, 1, 1, ...
    1,   0,   0,  0,  0, 0, 0, ...
    2,   2,   0,  0,  0, 0, 0, ...
    5,   3,   3,  0,  0, 0, 0, ...
   15,  16,   6,  4,  0, 0, 0, ...
   52,  65,  10, 10,  5, 0, 0, ...
  203, 336, 105, 20, 15, 6, 0, ...

Crossrefs

Columns k=0..3 give A000110, A052506, A354000, A354001.
Cf. A145460, A292892, A355610.

Programs

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

Formula

T(0,k) = 1 and T(n,k) = ((n-1)!/k!) * Sum_{j=k+1..n} (j/(j-k)!) * T(n-j,k)/(n-j)! for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(k!^j * (n-k*j)!).
Showing 1-5 of 5 results.

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

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