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.

A352069 Expansion of e.g.f. 1 / (1 + log(1 - 3*x) / 3).

Original entry on oeis.org

1, 1, 5, 42, 492, 7374, 134478, 2887128, 71281656, 1988802720, 61860849552, 2121993490176, 79566300371952, 3237181141173264, 142019158472311248, 6682603650677875584, 335698708873243355136, 17930674324049810882688, 1014685181110897126616448, 60641642160287342580586752
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 02 2022

Keywords

Crossrefs

Cf. A007840, A032031, A087674, A227917, A255927, A352071.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 + Log[1 - 3 x]/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! (-3)^(n - k), {k, 0, n}], {n, 0, 19}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-3*x)/3))) \\ Michel Marcus, Mar 02 2022

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-3)^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * 3^(k-1) * a(n-k).
a(n) ~ n! * 3^(n+1) * exp(3*n) / (exp(3) - 1)^(n+1). - Vaclav Kotesovec, Mar 03 2022

A352071 Expansion of e.g.f. 1 / (1 + log(1 - 4*x) / 4).

Original entry on oeis.org

1, 1, 6, 62, 904, 16984, 390128, 10586736, 331267200, 11738697600, 464539452672, 20302660659456, 971106358760448, 50452643588275200, 2829000818124208128, 170271405502300207104, 10948525752699316371456, 748994717201835804033024, 54315931193865932254543872
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 02 2022

Keywords

Crossrefs

Cf. A007840, A047053, A210246, A227917, A326324, A352069.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(1 + Log[1 - 4 x]/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! (-4)^(n - k), {k, 0, n}], {n, 0, 18}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-4*x)/4))) \\ Michel Marcus, Mar 02 2022

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-4)^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * 4^(k-1) * a(n-k).
a(n) ~ n! * 4^(n+1) * exp(4*n) / (exp(4) - 1)^(n+1). - Vaclav Kotesovec, Mar 03 2022

A352074 a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-n)^(n-k).

Original entry on oeis.org

1, 1, 4, 42, 904, 34070, 2019888, 174588120, 20804747136, 3276218158560, 659664288364800, 165425062846302336, 50574549124825998336, 18520126461205806360144, 8003819275469728355033088, 4031020344281171589447408000, 2340375822778055527109749211136
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 02 2022

Keywords

Links

Crossrefs

Cf. A007840, A092985, A094420, A227917, A317171, A317172, A331690, A352069, A352071.
Cf. A081048.

Programs

  • Mathematica
    Unprotect[Power]; 0^0 = 1; Table[Sum[StirlingS1[n, k] k! (-n)^(n - k), {k, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! SeriesCoefficient[1/(1 + Log[1 - n x]/n), {x, 0, n}], {n, 1, 16}]]
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 1)*k!*(-n)^(n-k)); \\ Michel Marcus, Mar 02 2022

Formula

a(n) = n! * [x^n] 1 / (1 + log(1 - n*x) / n) for n > 0.
a(n) ~ n! * n^(n-2) * (1 + 2*log(n)/n). - Vaclav Kotesovec, Mar 03 2022

A354237 Expansion of e.g.f. 1 / (1 - log(1 + 2*x) / 2).

Original entry on oeis.org

1, 1, 0, 2, -8, 64, -592, 6768, -90624, 1395840, -24292608, 471453696, -10094066688, 236340378624, -6007053852672, 164713554069504, -4846361933021184, 152300800682754048, -5091189648734748672, 180386551596145508352, -6752521487083688165376
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 06 2022

Keywords

Crossrefs

Cf. A006252, A088500, A088501, A122704, A155585, A227917, A354750, A354751.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 - Log[1 + 2 x]/2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 2^(n - k), {k, 0, n}], {n, 0, 20}]
  • PARI
    my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+2*x)/2))) \\ Michel Marcus, Jun 06 2022

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * 2^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * (-2)^(k-1) * a(n-k).
a(n) ~ n! * (-1)^(n+1) * 2^(n+1) / (n * log(n)^2) * (1 - (4 + 2*gamma)/log(n) + (12 + 12*gamma + 3*gamma^2 - Pi^2/2)/log(n)^2 + (2*Pi^2*gamma - 32 + 4*Pi^2 - 24*gamma^2 - 8*zeta(3) - 4*gamma^3 - 48*gamma)/log(n)^3 + (80 - 20*Pi^2*gamma + 40*zeta(3)*gamma - 5*Pi^2*gamma^2 + 160*gamma + 5*gamma^4 + 80*zeta(3) + 40*gamma^3 + Pi^4/12 - 20*Pi^2 + 120*gamma^2)/log(n)^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 06 2022

A367851 Expansion of e.g.f. 1/(1 - x + log(1 - 2*x)/2).

Original entry on oeis.org

1, 2, 10, 80, 872, 11984, 198416, 3840192, 85031040, 2119385856, 58714881792, 1789646610432, 59515302478848, 2144299161348096, 83204666280609792, 3459286210445942784, 153413140701637804032, 7228914528043587796992, 360670654712328998289408
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A052820, A367852, A367853.
Cf. A227917, A355110, A367845.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 2^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

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

A383170 Expansion of e.g.f. -log(1 + log(1 - 2*x)/2).

Original entry on oeis.org

0, 1, 3, 16, 122, 1208, 14704, 212336, 3547984, 67337728, 1430990976, 33664165632, 868592478720, 24390846882816, 740570519159808, 24177326011834368, 844599686386919424, 31438092340685144064, 1242230898248798896128, 51933512200489564962816, 2290351520336982559358976
Offset: 0

Views

Author

Seiichi Manyama, Apr 18 2025

Keywords

Crossrefs

Cf. A383171, A383172.
Cf. A003713, A227917.

Programs

  • PARI
    a(n) = sum(k=1, n, 2^(n-k)*abs(stirling(n, k, 1)*stirling(k, 1, 1)));

Formula

a(n) = Sum_{k=1..n} 2^(n-k) * |Stirling1(n,k) * Stirling1(k,1)|.
a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 1/2) / (exp(1) - exp(-1))^n. - Vaclav Kotesovec, Apr 18 2025

A368449 Expansion of e.g.f. exp(x) / (1 + log(1 - 2*x)/2).

Original entry on oeis.org

1, 2, 7, 42, 365, 4090, 55699, 890722, 16341849, 338128594, 7786397471, 197460558394, 5467207989957, 164085022299146, 5305738076252587, 183876885720455218, 6798985094507177137, 267160159254659407650, 11116956337133269707319, 488348854052875260086474
Offset: 0

Views

Author

Seiichi Manyama, Dec 24 2023

Keywords

Crossrefs

Cf. A291979, A368450.
Cf. A227917.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=1, i, 2^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(n) = 1 + Sum_{k=1..n} 2^(k-1) * (k-1)! * binomial(n,k) * a(n-k).

A377737 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 - log(1-2*x) / 2) ).

Original entry on oeis.org

1, 1, 4, 32, 392, 6504, 136464, 3466224, 103425664, 3546396288, 137423600640, 5939224680960, 283254408582144, 14777481937449984, 837175325044101120, 51182161648716349440, 3358765321328869539840, 235492308312669671424000, 17568539556367396687183872
Offset: 0

Views

Author

Seiichi Manyama, Nov 08 2024

Keywords

Links

Crossrefs

Cf. A138013, A377803.
Cf. A201595, A227917, A370938, A377789.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} 2^(n-k) * |Stirling1(n,k)|/(n-k+1)!.

A371299 E.g.f. satisfies A(x) = 1/(1 + log(1 - 2*x*A(x)^2) / 2).

Original entry on oeis.org

1, 1, 8, 128, 3128, 103464, 4327376, 219132416, 13037220864, 891482661120, 68898795919872, 5939542370104320, 565085390314014720, 58814874313859198976, 6647869870080852418560, 810941992663677532667904, 106188636284967568536207360, 14856670240947944840012857344
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2024

Keywords

Crossrefs

Cf. A227917, A370938.
Cf. A371297.

Programs

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

Formula

a(n) = (1/(2*n+1)!) * Sum_{k=0..n} 2^(n-k) * (2*n+k)! * |Stirling1(n,k)|.
Showing 1-9 of 9 results.

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

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