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.

A353999 Expansion of e.g.f. 1/(1 - x^3/6 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 20, 35, 1176, 10164, 58920, 277365, 3363580, 47567806, 519759604, 4591587455, 51017687280, 786120055400, 12187597925136, 165128862881769, 2261843835692340, 36940778814100210, 678763188831800380, 12143893591131411571, 211404290379223149384
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Crossrefs

Cf. A052848, A353998.
Cf. A000292, A351506, A354001.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3/6*(exp(x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!/6*sum(j=4, i, 1/(j-3)!*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, k!*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

Formula

a(0) = 1; a(n) = n!/6 * Sum_{k=4..n} 1/(k-3)! * a(n-k)/(n-k)! = binomial(n,3) * Sum_{k=4..n} binomial(n-3,k-3) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/4)} k! * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).

A354000 Expansion of e.g.f. exp(x^2/2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 3, 6, 10, 105, 651, 2968, 18936, 152505, 1164295, 9109056, 80012868, 756041377, 7387199925, 75535791360, 816560002576, 9254683835073, 109135702334619, 1338613513677280, 17079079303721820, 226148006163689841, 3100114305453613393, 43935964285680790368
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Links

Crossrefs

Cf. A052506, A354001.
Cf. A353998, A351492.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/2*(exp(x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/2*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(2^k*(n-2*k)!));

Formula

a(0) = 1; a(n) = ((n-1)!/2) * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(2^k * (n-2*k)!).

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)

A356952 E.g.f. satisfies log(A(x)) = x^3/6 * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 20, 35, 1736, 15204, 88320, 415965, 7632460, 121801966, 1368227224, 12184672955, 176889193040, 3490851044360, 59703361471296, 837948141904569, 13407228541467540, 283596013866706450, 6226093732482731800, 121326684752194084471
Offset: 0

Views

Author

Seiichi Manyama, Sep 06 2022

Keywords

Links

Crossrefs

Cf. A052880, A355843, A356951.
Cf. A000272, A354001, A356753, A356950.

Programs

  • Mathematica
    nmax = 23; A[_] = 1;
Do[A[x_] = Exp[x^3/6*(Exp[x] - 1)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, (k+1)^(k-1)*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^3/6*(exp(x)-1))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3/6*(1-exp(x))))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3/6*(1-exp(x)))/(x^3/6*(1-exp(x)))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^3/6 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^3/6 * (1 - exp(x))) ).
E.g.f.: A(x) = LambertW(x^3/6 * (1 - exp(x)))/(x^3/6 * (1 - exp(x))).

A368174 Expansion of e.g.f. -log(1 - x^3/6 * (exp(x) - 1)).

Original entry on oeis.org

0, 0, 0, 0, 4, 10, 20, 35, 616, 5124, 29520, 138765, 1312300, 16576846, 175795984, 1539037955, 15687832720, 216382727240, 3170822906976, 42007311638169, 553841577209940, 8435274815148370, 145708900713412960, 2517047758252082671, 42575155321545439384
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2023

Keywords

Comments

This sequence is different from A354001.

Links

Crossrefs

Cf. A052858, A368173.
Cf. A353999, A354001.

Programs

  • PARI
    a(n) = n!*sum(k=1, n\4, (k-1)!*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=1..floor(n/4)} (k-1)! * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).
a(0) = a(1) = a(2) = a(3) = 0; a(n) = binomial(n,3) + Sum_{k=4..n-1} binomial(k,3) * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Jan 22 2025

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 1-6 of 6 results.

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