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 17 results. Next

A089148 Expansion of e.g.f.: 1/(exp(x) - x).

Original entry on oeis.org

1, 0, -1, -1, 5, 19, -41, -519, -183, 19223, 73451, -847067, -8554547, 32488611, 977198559, 1325135969, -116987762287, -860498433233, 13730866757587, 243612350234973, -1120827248102379, -62079344419449925, -185852602587850681, 15185914155303053209
Offset: 0

Views

Author

Wouter Meeussen, Dec 06 2003

Keywords

Comments

INVERTi transform of [1, 1, 1/2, 1/6, 1/24, 1/120, ...] = [1, 0, -1/2, 1/6, 5/24, -19/120, -41/720, 519/5040, -183/40320, -19223/362880, ...]. - Gary W. Adamson, Oct 08 2008

Links

Crossrefs

Cf. A006153, A302397, A089087.

Programs

  • Maple
    b:= proc(n) b(n):= -`if` (n<0, 1, add(b(n-i)/(i-1)!, i=1..n+1)) end:
a:= n-> (-1)^n*n!*b(n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 29 2013
  • Mathematica
    a = CoefficientList[Series[1/( E^ x - x), {x, 0, 30}], x]; Table[(n - 1)! *a[[n]], {n, 1, Length[a]}] (* Roger L. Bagula and Gary W. Adamson, Oct 06 2008 *)
With[{nn=30},CoefficientList[Series[1/(Exp[x]-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 03 2017 *)
  • Maxima
    a(n):=sum(sum(k!*binomial(n,l)*(-1)^(k-l)*stirling2(n-l,k-l), l,0,k), k,0,n); /* Vladimir Kruchinin, May 29 2013 */
  • Maxima
    a(n):=n!*sum((-n-1+k)^k/k!,k,0,n); /* Tani Akinari, Mar 26 2023 */
  • PARI
    x='x+O('x^66); Vec(serlaplace(1/(exp(x)-x))) \\ Joerg Arndt, May 29 2013
  • Sage
    def A089148_list(len):
    f, R, C = 1, [], [1]+[0]*len
    for n in (1..len):
        for k in range(n, 0, -1):
            C[k] = C[k-1]*(1/(k-1) if k>1 else 1)
        C[0] = -sum((-1)^k*C[k] for k in (1..n))
        R.append(C[0]*f)
        f *= n
    return R
print(A089148_list(24)) # Peter Luschny, Feb 21 2016

Formula

E.g.f.: -(1+1/(G(0)-1))/x where G(k) = 1 - (k+1)/(1 - x/(x + (k+1)^2/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 07 2012
a(n) = Sum_{k=0..n} Sum_{m=0..k} k!*binomial(n,m)*(-1)^(k-m)*Stirling2(n-m,k-m). - Vladimir Kruchinin, May 29 2013
Lim sup n->oo |a(n)/n!|^(1/n) = 1/abs(LambertW(-1)) = 0.727507111152... - Vaclav Kotesovec, Aug 13 2013
a(0) = 1; a(n) = -Sum_{k=2..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020
a(n) = n!*Sum_{k=0..n} (-n-1+k)^k/k!. - Tani Akinari, Mar 25 2023
a(n) = Sum_{k=0..n} A089087(n, k). - Peter Luschny, Mar 25 2023

A352252 Expansion of e.g.f. 1 / (1 - x * cos(x)).

Original entry on oeis.org

1, 1, 2, 3, 0, -55, -480, -3157, -15232, -16623, 898560, 16316179, 194574336, 1666248025, 5418649600, -170157839685, -5164467978240, -92955464490463, -1188910801354752, -7329026447550685, 157257042777866240, 7516793832172469481, 187200588993188069376
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Links

Crossrefs

Cf. A000111, A007289, A009189, A094088, A205571, A302397, A352250, A352251.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - x Cos[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^k Binomial[n, 2 k + 1] (2 k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 22}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(1 / (1 - x * cos(x)))) \\ Michel Marcus, Mar 10 2022
  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*I^(n-k)*a185951(n, k)); \\ Seiichi Manyama, Feb 17 2025

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * (2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 17 2025

A305990 Expansion of e.g.f.: (1+x) / (exp(-x) - x).

Original entry on oeis.org

1, 3, 11, 58, 409, 3606, 38149, 470856, 6641793, 105398650, 1858413061, 36044759796, 762659322385, 17481598316742, 431535346662645, 11413394655983536, 321989729198400385, 9651573930139850610, 306321759739045148293, 10262156907184058219340
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 16 2018

Keywords

Crossrefs

Cf. A305133, A006153, A009306, A009444, A072597, A089148, A302397.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1+x)/(E^(-x)-x), {x, 0, nmax}], x] * Range[0, nmax]!
a={1};For[n=1,n<20,n++,AppendTo[a,Sum[(n!)*((n-k+1)^(k-1))*(n+1)/(k!),{k,0,n+1}]]]; a (* Detlef Meya, Sep 05 2023 *)

Formula

a(n) ~ n! / LambertW(1)^(n+1).
a(n) = (-1)^n * A009444(n+1).
a(n) = Sum_{k=0..n+1} (n+1)!*(n-k+1)^(k-1)/k! for n > 0. - Detlef Meya, Sep 05 2023

A351776 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * (n-j)^j/j!.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 4, 3, 0, 1, -4, 12, -6, -4, 0, 1, -5, 24, -63, -8, -25, 0, 1, -6, 40, -204, 420, 150, 114, 0, 1, -7, 60, -465, 2288, -3435, -972, 287, 0, 1, -8, 84, -882, 7180, -32020, 33462, 3682, -4152, 0, 1, -9, 112, -1491, 17256, -138525, 537576, -379155, 6256, 1647, 0
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Examples

			Square array begins:
  1,   1,   1,     1,      1,       1, ...
  0,  -1,  -2,    -3,     -4,      -5, ...
  0,   0,   4,    12,     24,      40, ...
  0,   3,  -6,   -63,   -204,    -465, ...
  0,  -4,  -8,   420,   2288,    7180, ...
  0, -25, 150, -3435, -32020, -138525, ...

Crossrefs

Columns k=0..3 give A000007, A302397, A351777, A351778.
Main diagonal gives A351779.
Cf. A292861, A351761.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n, (-k)^(n-j)*(n-j)^j/j!);
  • PARI
    T(n, k) = if(n==0, 1, -k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));

Formula

E.g.f. of column k: 1/(1 + k*x*exp(x)).
T(0,k) = 1 and T(n,k) = -k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

A351791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (-k * (n-j))^j/j!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 0, 6, 1, 1, -2, -3, 24, 1, 1, -4, -6, -4, 120, 1, 1, -6, -3, 40, 25, 720, 1, 1, -8, 6, 132, 120, 114, 5040, 1, 1, -10, 21, 248, -375, -1872, -287, 40320, 1, 1, -12, 42, 364, -2120, -8298, -3920, -4152, 362880, 1, 1, -14, 69, 456, -5655, -12144, 86121, 155776, -1647, 3628800
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Examples

			Square array begins:
    1,  1,   1,    1,     1,     1, ...
    1,  1,   1,    1,     1,     1, ...
    2,  0,  -2,   -4,    -6,    -8, ...
    6, -3,  -6,   -3,     6,    21, ...
   24, -4,  40,  132,   248,   364, ...
  120, 25, 120, -375, -2120, -5655, ...

Crossrefs

Columns k=0..4 give A000142, (-1)^n * A302397(n), A336959, A351792, A351793.
Main diagonal gives (-1)^n * A302398(n).
Cf. A351776, A351790.

Programs

  • Mathematica
    T[n_, k_] := n!*(1 + Sum[(-k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 19 2022 *)
  • PARI
    T(n, k) = n!*sum(j=0, n, (-k*(n-j))^j/j!);
  • PARI
    T(n, k) = if(n==0, 1, n*sum(j=0, n-1, (-k)^(n-1-j)*binomial(n-1, j)*T(j, k)));

Formula

E.g.f. of column k: 1/(1 - x*exp(-k*x)).
T(0,k) = 1 and T(n,k) = n * Sum_{j=0..n-1} (-k)^(n-1-j) * binomial(n-1,j) * T(j,k) for n > 0.

A368271 Expansion of e.g.f. exp(2*x) / (1 + x*exp(x)).

Original entry on oeis.org

1, 1, 0, -1, 4, 7, -74, 23, 2136, -7345, -77006, 712879, 2499124, -69799897, 88342398, 7311735143, -50617554896, -762825930977, 12821702643946, 56041362405119, -2956159258069044, 8447845572175031, 660257137187089270, -7376306690095890185
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Cf. A302397, A368272, A368273.
Cf. A368265, A368267.

Programs

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

Formula

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

A302398 a(n) = n! * [x^n] 1/(1 + x*exp(n*x)).

Original entry on oeis.org

1, -1, -2, 3, 248, 5655, 62064, -3516625, -376936064, -21890186577, -495165203200, 96687112380639, 20607024735783936, 2471270260977141767, 142697263160045590528, -25986252776953159328625, -11860424645318274482077696, -2719428501410438623907546529, -372732332273232481973818294272
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 07 2018

Keywords

Crossrefs

Cf. A134095, A235328, A302397.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(1 + x Exp[n x]), {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n! Sum[(-1)^(n - k) (n (n - k))^k/k!, {k, 0, n}], {n, 18}]]
Join[{1}, Table[Sum[(-1)^k k! (n k)^(n - k) Binomial[n, k], {k, 0, n}], {n, 18}]]

Formula

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

A332627 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * k! * k^n.

Original entry on oeis.org

1, 1, 6, 117, 4388, 266065, 23731314, 2923345621, 475364541672, 98623225721601, 25421365316232710, 7969388199705535141, 2985785305877403047820, 1317500933136749853197329, 676266417871227455138941242, 399516621958550611386236160405
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2020

Keywords

Links

Crossrefs

Cf. A000166, A120485, A134095, A302397, A319392, A332408.

Programs

  • Mathematica
    Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] k! k^n, {k, 0, n}], {n, 1, 15}]]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * k! * k^n); \\ Michel Marcus, Apr 24 2020
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x*exp(-x))^k))) \\ Seiichi Manyama, Feb 19 2022

Formula

G.f.: Sum_{k>=0} k! * k^k * x^k / (1 + k*x)^(k+1).
a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^n / (n-k)!.
a(n) ~ c * n! * n^n, where c = A072364 = exp(-exp(-1)). - Vaclav Kotesovec, Jul 10 2021
E.g.f.: Sum_{k>=0} (k*x*exp(-x))^k. - Seiichi Manyama, Feb 19 2022

A336959 E.g.f.: 1 / (1 - x * exp(-2*x)).

Original entry on oeis.org

1, 1, -2, -6, 40, 120, -1872, -3920, 155776, 56448, -19946240, 44799744, 3588719616, -21265587200, -850126505984, 9423227873280, 251457224998912, -4665150579572736, -88212028284665856, 2663461772025462784, 34353949630376181760, -1756678038088484388864
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 09 2020

Keywords

Crossrefs

Cf. A001787, A155585, A302397, A336950, A336958.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[1/(1 - x Exp[-2 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(-2 (n - k))^k/k!, {k, 0, n}], {n, 1, 21}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k (-2)^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

Formula

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

A352250 Expansion of e.g.f. 1 / (1 - x * sin(x)) (even powers only).

Original entry on oeis.org

1, 2, 20, 486, 21944, 1591210, 169207092, 24808395262, 4796420822384, 1182349445882706, 361939981107422060, 134705596642758848806, 59900689507397744253096, 31365504832631796986962426, 19102102945852191813235300004, 13387748268024668296590660222030
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Crossrefs

Cf. A000111, A000364, A009214, A205571, A210657, A302397, A352251, A352252.

Programs

  • Mathematica
    nmax = 30; Take[CoefficientList[Series[1/(1 - x Sin[x]), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = 2 Sum[(-1)^(k + 1) Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
  • PARI
    my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sin(x))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

a(0) = 1; a(n) = 2 * Sum_{k=1..n} (-1)^(k+1) * binomial(2*n,2*k) * k * a(n-k).
Showing 1-10 of 17 results. Next

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