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 1-9 of 9 results.

A353998 Expansion of e.g.f. 1/(1 - x^2/2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 3, 6, 10, 195, 1281, 5908, 68076, 758565, 6486535, 75598446, 1059484218, 13378016743, 185273328345, 2999003869800, 48665352612376, 816394913567433, 15110162148144267, 292156921946387170, 5805684093139498470, 122617308231635240331
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Crossrefs

Cf. A052848, A353999.
Cf. A351505, A354000.

Programs

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

Formula

a(0) = 1; a(n) = n!/2 * Sum_{k=3..n} 1/(k-2)! * a(n-k)/(n-k)! = binomial(n,2) * Sum_{k=3..n} binomial(n-2,k-2) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/3)} k! * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).
a(n) ~ 2 * n! / ((4 + 2*r + r^3) * r^n), where r = 1.043121496712693605897520269472163423276582653660720448... is the root of the equation (exp(r)-1)*r^2 = 2. - Vaclav Kotesovec, May 13 2022

A346889 Expansion of e.g.f. 1 / (1 - x^3 * exp(x) / 3!).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 40, 315, 2296, 15204, 117720, 1127445, 11531740, 120909646, 1370809804, 17111895255, 227853866800, 3182209445640, 47003318806896, 737325061500009, 12187616610231540, 210930852047426770, 3821604062633503300, 72479758506840597451
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 06 2021

Keywords

Links

Crossrefs

Column k=3 of A351703.
Cf. A006153, A346888, A346890, A346893.
Cf. A000292, A145453, A353999.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - x^3 Exp[x]/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 3] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^3*exp(x)/3!))) \\ Michel Marcus, Aug 06 2021
  • PARI
    a(n) = n!*sum(k=0, n\3, k^(n-3*k)/(6^k*(n-3*k)!)); \\ Seiichi Manyama, May 13 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * binomial(k,3) * a(n-k).
a(n) ~ n! / ((1 + LambertW(2^(1/3)/3^(2/3))) * 3^(n+1) * LambertW(2^(1/3)/3^(2/3))^n). - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-3*k)/(6^k * (n-3*k)!). - Seiichi Manyama, May 13 2022

A354001 Expansion of e.g.f. exp(x^3/6 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 20, 35, 616, 5124, 29520, 138765, 942700, 9369646, 91711984, 782281955, 6539493520, 62576274440, 693828386976, 7968383514969, 89851862221140, 1023732374445970, 12384993316732960, 160496534000858671, 2163244034675904664, 29653387436468336300
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Links

Crossrefs

Cf. A052506, A354000.
Cf. A351493, A353999.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x^3/6 (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 07 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(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-1)!/6*sum(j=4, i, j/(j-3)!*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

Formula

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

A358014 Expansion of e.g.f. 1/(1 - x^3 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 120, 210, 40656, 363384, 2117520, 9980190, 520250280, 9496208436, 109522054824, 982593614730, 28426015541280, 762523155318000, 14192088961120416, 204618562767970614, 4906638448867994040, 154037798077765359660, 4000484484370905087480
Offset: 0

Views

Author

Seiichi Manyama, Oct 24 2022

Keywords

Crossrefs

Cf. A052848, A358013.
Cf. A292891, A351504, A353999.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1-x^3 (Exp[x]-1)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 26 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3*(exp(x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*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)/(n-3*k)!);

Formula

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

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

A354314 Expansion of e.g.f. 1/(1 - x/3 * (exp(3 * x) - 1)).

Original entry on oeis.org

1, 0, 2, 9, 60, 495, 4986, 58401, 780984, 11749779, 196446870, 3612882933, 72484364052, 1575418827879, 36875093680530, 924769734574185, 24737895033896304, 703105981990977915, 21159355356941587470, 672148402091190649629, 22475238194908656800460
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A052848, A354313.
Cf. A288834, A328182, A353999, A354312.

Programs

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

Formula

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

A370992 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^3/6*(exp(x) - 1)) ).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 20, 35, 5656, 55524, 352920, 1801965, 85636540, 1762160686, 22992890284, 232001269955, 6581012518640, 197506018950920, 4224661065644016, 69931313468126169, 1757395269147356340, 60785516594782517650, 1818493252905482003620
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A370988, A370991.
Cf. A353999.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^3/6*(exp(x)-1)))/x))
  • PARI
    a(n) = sum(k=0, n\4, (n+k)!*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!))/(n+1);

Formula

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

A355308 Expansion of e.g.f. -LambertW(x^3/6 * (1 - exp(x))).

Original entry on oeis.org

0, 0, 0, 0, 4, 10, 20, 35, 1176, 10164, 58920, 277365, 4472380, 69189406, 772011604, 6861855455, 95279504880, 1819310613800, 30768119885136, 430200439251369, 6770486332450740, 139958614722287410, 3033142442978720380, 58782387380290683571, 1138026666874389737544
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2022

Keywords

Links

Crossrefs

Cf. A048802, A355181, A357267.
Cf. A353999, A355180.

Programs

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

Formula

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

A355666 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k/k! * (exp(x) - 1)).

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 0, 2, 13, 1, 0, 0, 3, 75, 1, 0, 0, 3, 28, 541, 1, 0, 0, 0, 6, 125, 4683, 1, 0, 0, 0, 4, 10, 1146, 47293, 1, 0, 0, 0, 0, 10, 195, 8827, 545835, 1, 0, 0, 0, 0, 5, 20, 1281, 94200, 7087261, 1, 0, 0, 0, 0, 0, 15, 35, 5908, 1007001, 102247563, 1, 0, 0, 0, 0, 0, 6, 35, 1176, 68076, 12814390, 1622632573
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2022

Keywords

Examples

			Square array begins:
     1,    1,   1,  1,  1, 1, 1, ...
     1,    0,   0,  0,  0, 0, 0, ...
     3,    2,   0,  0,  0, 0, 0, ...
    13,    3,   3,  0,  0, 0, 0, ...
    75,   28,   6,  4,  0, 0, 0, ...
   541,  125,  10, 10,  5, 0, 0, ...
  4683, 1146, 195, 20, 15, 6, 0, ...

Crossrefs

Columns k=0..3 give A000670, A052848, A353998, A353999.
Cf. A351703, A355652.

Programs

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

Formula

T(0,k) = 1 and T(n,k) = binomial(n,k) * Sum_{j=k+1..n} binomial(n-k,j-k) * T(n-j,k) for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * Stirling2(n-k*j,j)/(k!^j * (n-k*j)!).
Showing 1-9 of 9 results.

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