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 1-10 of 10 results.

A319508 a(n) = n! * [x^n] 1/(1 + n - exp(x)*(exp(n*x) - 1)/(exp(x) - 1)).

Original entry on oeis.org

1, 1, 23, 1836, 361754, 143195025, 99986786773, 112625837135056, 191736660977760804, 469456525723134676365, 1589874326596159958849175, 7216642860485686755145923828, 42781019992428263086709058587150, 324097110833947198922869762652717041
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 21 2018

Keywords

Links

Crossrefs

Main diagonal of A320253.
Cf. A000670, A004700, A004701, A004702, A004703, A004704, A004705, A004706, A004707, A004708, A319509.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(1 + n - Exp[x] (Exp[n x] - 1)/(Exp[x] - 1)), {x, 0, n}], {n, 0, 13}]
  • PARI
    default(seriesprecision, 101); {a(n) = n!*polcoeff((1/(1+n-exp(x)*(exp(n*x)-1)/(exp(x)-1)) + O(x^(n+1))), n)};
for(n=0, 15, print1(a(n), ", ")) \\ G. C. Greubel, Oct 09 2018

Formula

a(n) = n! * [x^n] 1/(1 + n - exp(x) - exp(2*x) - exp(3*x) - ... - exp(n*x)).
a(n) ~ sqrt(2*Pi) * n^(3*n + 1/2) / (2^n * exp(n - 5/3)). - Vaclav Kotesovec, Oct 09 2018

A004700 Expansion of e.g.f. 1/(3 - exp(x) - exp(2*x)).

Original entry on oeis.org

1, 3, 23, 261, 3947, 74613, 1692563, 44794221, 1354849547, 46101247173, 1742977452803, 72487571292381, 3288697207653947, 161639067567489333, 8555659001848069043, 485203383272476257741, 29350999686572204663147, 1886474390059466622333093
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=2 of A320253.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(3-Exp(x)-Exp(2*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 09 2018
  • Maple
    seq(coeff(series(factorial(n)*(3-exp(x)-exp(2*x))^(-1),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 10 2018
  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(3-Exp[x]-Exp[2x]),{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Dec 04 2011 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(3-sum(k=1,2, exp(k*x))))) \\ G. C. Greubel, Oct 09 2018

Formula

G.f.: 1/(3 - E(0)), where E(k)= 1 + 2^k/(1 - x/(x + 2^k*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 21 2013
a(n) ~ 2*n!/((13-sqrt(13))*(log((sqrt(13)-1)/2))^(n+1)). - Vaclav Kotesovec, Aug 13 2013
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (2^k + 1) * a(n-k). - Ilya Gutkovskiy, Jan 15 2020

A004701 Expansion of e.g.f. 1/(4 - exp(x) - exp(2*x) - exp(3*x)).

Original entry on oeis.org

1, 6, 86, 1836, 52250, 1858716, 79345346, 3951633636, 224917803770, 14402023566156, 1024662142371506, 80191908540219636, 6846505625682597290, 633241684193651067996, 63074628985206471485666, 6731364953866743063784836
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=3 of A320253.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(4-Exp(x)-Exp(2*x)-Exp(3*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 09 2018
  • Maple
    seq(coeff(series(factorial(n)*(4-exp(x)-exp(2*x)-exp(3*x))^(-1),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Oct 10 2018
  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(4-Exp[x]-Exp[2*x]-Exp[3*x]),{x,0,nn}],x] Range[0,nn]!] (* Vincenzo Librandi, Jun 14 2012 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(4-sum(k=1,3, exp(k*x))))) \\ G. C. Greubel, Oct 09 2018

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (1 + 2^k + 3^k) * a(n-k). - Ilya Gutkovskiy, Jan 15 2020

A004702 Expansion of e.g.f. 1/(5 - exp(x) - exp(2*x) - exp(3*x) - exp(4*x)).

Original entry on oeis.org

1, 10, 230, 7900, 361754, 20706700, 1422295490, 113976565300, 10438383399674, 1075482742196860, 123120717545481650, 15504276864309866500, 2129906079562267271594, 316979734672375940684620, 50802750419531400066083810
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=4 of A320253.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(5-Exp(x)-Exp(2*x)-Exp(3*x)-Exp(4*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 09 2018
  • Maple
    seq(coeff(series(factorial(n)*(5-exp(x)-exp(2*x)-exp(3*x)-exp(4*x))^(-1),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Oct 10 2018
  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(5-Exp[x]-Exp[2*x]-Exp[3*x]-Exp[4*x]),{x,0,nn}],x] Range[0,nn]!] (* Vincenzo Librandi, Jun 14 2012 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(5-sum(k=1,4, exp(k*x))))) \\ G. C. Greubel, Oct 09 2018

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (1 + 2^k + 3^k + 4^k) * a(n-k). - Ilya Gutkovskiy, Jan 15 2020

A004703 Expansion of e.g.f. 1/(6-exp(x)-exp(2*x)-exp(3*x)-exp(4*x)-exp(5*x)).

Original entry on oeis.org

1, 15, 505, 25425, 1706629, 143195025, 14417768365, 1693616001225, 227365098508549, 34338804652192545, 5762408433135346525, 1063691250037869293625, 214198140845740727508469, 46728077502266943919186065
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=5 of A320253.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(6-Exp(x)-Exp(2*x)-Exp(3*x)-Exp(4*x)-Exp(5*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 09 2018
  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(6-Exp[x]-Exp[2*x]-Exp[3*x] -Exp[4*x]-Exp[5*x]),{x,0,nn}],x] Range[0,nn]!] (* Vincenzo Librandi, Jun 14 2012 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(6-sum(k=1,5, exp(k*x))))) \\ G. C. Greubel, Oct 09 2018

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (1 + 2^k + ... + 5^k) * a(n-k). - Ilya Gutkovskiy, Jan 15 2020

A004704 Expansion of e.g.f. 1/(7- Sum_{k=1..6} exp(k*x)).

Original entry on oeis.org

1, 21, 973, 67473, 6238309, 720964881, 99986786773, 16177741934193, 2991473373828709, 622307309978695761, 143840821212045590773, 36572284571798550251313, 10144031468802588684994309, 3048113900510603294243693841
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=6 of A320253.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(7-Exp(x)-Exp(2*x)-Exp(3*x)-Exp(4*x)-Exp(5*x)-Exp(6*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 09 2018
  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(7-Exp[x]-Exp[2*x]-Exp[3*x]-Exp[4*x]-Exp[5*x]-Exp[6*x]),{x,0,nn}],x] Range[0,nn]!] (* Vincenzo Librandi, Jun 14 2012 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(7-sum(k=1,6, exp(k*x))))) \\ G. C. Greubel, Oct 09 2018

Formula

Equals expansion of e.g.f. 1/(7-exp(x)-exp(2*x)-exp(3*x)-exp(4*x)-exp(5*x)-exp(6*x)).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (1 + 2^k + ... + 6^k) * a(n-k). - Ilya Gutkovskiy, Jan 15 2020

A004705 Expansion of e.g.f. 1/(8 - Sum_{k=1..7} exp(k*x)).

Original entry on oeis.org

1, 28, 1708, 156016, 19000996, 2892636208, 528436162708, 112625837135056, 27433137537640996, 7517361789179684848, 2288826715171726889908, 766572192067000875962896, 280079787805796188648857796, 110859415083883527695265783088
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=7 of A320253.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(8-Exp(x)-Exp(2*x)-Exp(3*x)-Exp(4*x)-Exp(5*x)-Exp(6*x)-Exp(7*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 09 2018
  • Mathematica
    With[{nn=200},CoefficientList[Series[1/(8-Exp[x]-Exp[2*x]-Exp[3*x]-Exp[4*x]-Exp[5*x]-Exp[6*x]-Exp[7*x]),{x,0,nn}],x] Range[0,nn]!] (* Vincenzo Librandi, Jun 15 2012 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(8-sum(k=1,7, exp(k*x))))) \\ G. C. Greubel, Oct 09 2018

Formula

Equals expansion of e.g.f. 1/(8-exp(x)-exp(2*x)-exp(3*x)-exp(4*x)-exp(5*x)-exp(6*x)-exp(7*x)).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (1 + 2^k + ... + 7^k) * a(n-k). - Ilya Gutkovskiy, Jan 15 2020

A004706 Expansion of e.g.f. 1/(9 - Sum_{k=1..8} exp(k*x)).

Original entry on oeis.org

1, 36, 2796, 325296, 50460324, 9784339056, 2276639188116, 618021679767696, 191736660977760804, 66920493102763469616, 25951985825417984806836, 11070691364651231290738896, 5151900329218737241490290884
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=8 of A320253.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(9-Exp(x)-Exp(2*x)-Exp(3*x)-Exp(4*x)-Exp(5*x)-Exp(6*x)-Exp(7*x)-Exp(8*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 09 2018
  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(9-Exp[x]-Exp[2*x]-Exp[3*x]-Exp[4*x]-Exp[5*x]-Exp[6*x]-Exp[7*x]-Exp[8*x]),{x,0,nn}],x] Range[0,nn]!] (* Vincenzo Librandi, Jun 15 2012 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(1/(9-sum(k=1,8, exp(k*x))))) \\ G. C. Greubel, Oct 09 2018

Formula

Equals expansion of e.g.f. 1/(9-exp(x)-exp(2*x)-exp(3*x)-exp(4*x)-exp(5*x)-exp(6*x)-exp(7*x)-exp(8*x)).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (1 + 2^k + ... + 8^k) * a(n-k). - Ilya Gutkovskiy, Jan 15 2020

A355427 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - Sum_{j=1..k} (exp(j*x) - 1)/j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 11, 13, 0, 1, 4, 24, 89, 75, 0, 1, 5, 42, 284, 959, 541, 0, 1, 6, 65, 654, 4476, 12917, 4683, 0, 1, 7, 93, 1255, 13564, 88178, 208781, 47293, 0, 1, 8, 126, 2143, 32275, 351634, 2084564, 3937019, 545835, 0
Offset: 0

Views

Author

Seiichi Manyama, Jul 01 2022

Keywords

Examples

			Square array begins:
  1,   1,     1,     1,      1,       1, ...
  0,   1,     2,     3,      4,       5, ...
  0,   3,    11,    24,     42,      65, ...
  0,  13,    89,   284,    654,    1255, ...
  0,  75,   959,  4476,  13564,   32275, ...
  0, 541, 12917, 88178, 351634, 1037479, ...

Crossrefs

Columns k=0..3 give A000007, A000670, A355425, A355426.
Main diagonal gives A355428.
Cf. A306024, A320253.

Formula

T(0,k) = 1 and T(n,k) = Sum_{i=1..n} (Sum_{j=1..k} j^(i-1)) * binomial(n,i) * T(n-i,k) for n > 0.

A355423 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} (exp(j*x) - 1)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 6, 14, 5, 0, 1, 10, 50, 81, 15, 0, 1, 15, 130, 504, 551, 52, 0, 1, 21, 280, 2000, 5870, 4266, 203, 0, 1, 28, 532, 6075, 35054, 76872, 36803, 877, 0, 1, 36, 924, 15435, 148429, 684000, 1111646, 348543, 4140, 0
Offset: 0

Views

Author

Seiichi Manyama, Jul 01 2022

Keywords

Examples

			Square array begins:
  1,  1,    1,     1,      1,       1, ...
  0,  1,    3,     6,     10,      15, ...
  0,  2,   14,    50,    130,     280, ...
  0,  5,   81,   504,   2000,    6075, ...
  0, 15,  551,  5870,  35054,  148429, ...
  0, 52, 4266, 76872, 684000, 4004100, ...

Crossrefs

Columns k=0-4 give: A000007, A000110, A355291, A355421, A355422.
Main diagonal gives A320288.
Cf. A306024, A320253.

Formula

T(0,k) = 1 and T(n,k) = Sum_{i=1..n} (Sum_{j=1..k} j^i) * binomial(n-1,i-1) * T(n-i,k) for n > 0.
Showing 1-10 of 10 results.

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

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