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 1-8 of 8 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)!).

A346888 Expansion of e.g.f. 1 / (1 - x^2 * exp(x) / 2).

Original entry on oeis.org

1, 0, 1, 3, 12, 70, 465, 3591, 31948, 319068, 3539385, 43205635, 575312826, 8298867798, 128921967265, 2145837600375, 38097353658120, 718657756980376, 14354000800751313, 302625047150614179, 6716038666999745710, 156498725047355717250, 3820426102008414736761
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 06 2021

Keywords

Links

Crossrefs

Column k=2 of A351703.
Cf. A006153, A346889, A346890, A346893.
Cf. A000217, A133189, A308946, A353998.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - x^2 Exp[x]/2), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^2*exp(x)/2))) \\ Michel Marcus, Aug 06 2021
  • PARI
    a(n) = n!*sum(k=0, n\2, k^(n-2*k)/(2^k*(n-2*k)!)); \\ Seiichi Manyama, May 13 2022

Formula

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

A358013 Expansion of e.g.f. 1/(1 - x^2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 750, 5082, 23576, 453672, 5755770, 50894030, 841270452, 14694142476, 201442729670, 3552604015170, 73814245552560, 1369932831933392, 27860865121662066, 655240785723048726, 15052226249248287500, 357713461766745539700, 9416426612423343023742
Offset: 0

Views

Author

Seiichi Manyama, Oct 24 2022

Keywords

Crossrefs

Cf. A052848, A358014.
Cf. A240989, A351503, A353998.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2*(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=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)/(n-2*k)!);

Formula

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

A368173 Expansion of e.g.f. -log(1 - x^2/2 * (exp(x) - 1)).

Original entry on oeis.org

0, 0, 0, 3, 6, 10, 105, 651, 2968, 26496, 265905, 2203795, 22830456, 288661308, 3476579197, 44960585775, 671394654960, 10329701480416, 164573071219233, 2865785889662019, 52647629639499280, 1000194250108913580, 20125846165307543661, 426789766980101676943
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2023

Keywords

Links

Crossrefs

Cf. A052858, A368174.
Cf. A353998.

Programs

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

Formula

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

A354313 Expansion of e.g.f. 1/(1 - x/2 * (exp(2 * x) - 1)).

Original entry on oeis.org

1, 0, 2, 6, 40, 280, 2496, 25424, 297984, 3920256, 57349120, 922611712, 16193375232, 307896882176, 6304666798080, 138318662000640, 3236895083167744, 80483201605795840, 2118875812456366080, 58882581280649117696, 1722441885524719042560
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A052848, A354314.
Cf. A216794, A353998, A354311.

Programs

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

Formula

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

A370991 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^2/2*(exp(x) - 1)) ).

Original entry on oeis.org

1, 0, 0, 3, 6, 10, 735, 5691, 29428, 1122696, 18159165, 190810675, 5768268726, 143497346928, 2479363382587, 73013461310895, 2336253676913640, 58015822633914736, 1850758447642034553, 69357415099500398979, 2252468410247071488970
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A370988, A370992.
Cf. A353998.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (n+k)! * Stirling2(n-2*k,k)/(2^k * (n-2*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 1-8 of 8 results.

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

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