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.

Previous Showing 11-20 of 20 results.

A277485 E.g.f.: -exp(2*x)*LambertW(-x).

Original entry on oeis.org

0, 1, 6, 33, 216, 1865, 21228, 303765, 5222864, 104540337, 2383558740, 60933722069, 1725392415288, 53590463856345, 1811281159509500, 66172416761172885, 2598298697830360992, 109116931783034360801, 4880122696811960470692, 231565260558289051906965
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 17 2016

Keywords

Links

Crossrefs

Cf. A000169, A069856, A086331, A277457, A277473, A277474.

Programs

  • Mathematica
    CoefficientList[Series[-Exp[2*x]*LambertW[-x], {x, 0, 20}], x]*Range[0, 20]!
Table[Sum[Binomial[n, m]*Sum[Binomial[m, k]*k^(k-1), {k, 1, m}], {m, 1, n}], {n, 0, 20}]
  • PARI
    x='x+O('x^50); concat([0], Vec(serlaplace(- exp(2*x)*lambertw(-x) ))) \\ G. C. Greubel, Nov 08 2017

Formula

a(n) = Sum_{m=1..n} (binomial(n,m) * Sum_{k=1..m} binomial(m,k)*k^(k-1)).
a(n) ~ exp(2*exp(-1)) * n^(n-1).

A350454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, none of which are isolated; triangle T(n,k), n >= 0, 0 <= k <= n/2, read by rows.

Original entry on oeis.org

1, 0, 1, 2, 8, 9, 81, 76, 12, 1024, 875, 180, 15625, 12606, 2910, 120, 279936, 217217, 53550, 3780, 5764801, 4348856, 1118936, 102480, 1680, 134217728, 99111735, 26280072, 2817360, 90720, 3486784401, 2532027610, 686569050, 81864720, 3729600, 30240
Offset: 0

Views

Author

Alois P. Heinz, Dec 31 2021

Keywords

Examples

			Triangle T(n,k) begins:
           1;
           0;
           1,          2;
           8,          9;
          81,         76,        12;
        1024,        875,       180;
       15625,      12606,      2910,      120;
      279936,     217217,     53550,     3780;
     5764801,    4348856,   1118936,   102480,    1680;
   134217728,   99111735,  26280072,  2817360,   90720;
  3486784401, 2532027610, 686569050, 81864720, 3729600, 30240;
  ...

Links

Crossrefs

Column k=0 gives A065440.
Row sums give |A069856|.
T(2n,n) gives A001813.
Cf. A349454.

Programs

  • Maple
    c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
t:= proc(n) option remember; n^(n-1) end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-i)*
      binomial(n-1, i-1)*(c(i)+`if`(i=1, 0, x*t(i))), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..12);
# second Maple program:
egf := k-> exp(LambertW(-x))*(-x-LambertW(-x))^k/((1+LambertW(-x))*k!):
A350454 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A350454(n, k), k=0..n/2)), n=0..9); # Mélika Tebni, Nov 22 2022
  • Mathematica
    c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
t[n_] := t[n] = n^(n - 1);
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - i]*
     Binomial[n - 1, i - 1]*(c[i] + If[i == 1, 0, x*t[i]]), {i, 1, n}]]];
T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 06 2022, after Alois P. Heinz *)

Formula

E.g.f. column k: exp(W(-x))*(-x - W(-x))^k / ((1 + W(-x))*k!), W(x) the Lambert W-function. - Mélika Tebni, Nov 22 2022
From Mélika Tebni, Dec 22 2022: (Start)
For n > 1, T(n,1) = n*A045531(n-1).
Sum_{k=0..n} (-1)^(n-k)*T(n+k,k) = 2^n.
Sum_{k=0..n} (-1)^(n-k)*T(n+k,k)/(n+k-1) = 1/n, for n > 1. (End)

A362856 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-k)^(n-j) * j^j * binomial(n,j).

Original entry on oeis.org

1, 1, 1, 1, 0, 4, 1, -1, 3, 27, 1, -2, 4, 17, 256, 1, -3, 7, 7, 169, 3125, 1, -4, 12, -9, 120, 2079, 46656, 1, -5, 19, -37, 121, 1373, 31261, 823543, 1, -6, 28, -83, 208, 797, 21028, 554483, 16777216, 1, -7, 39, -153, 441, 21, 14517, 373931, 11336753, 387420489
Offset: 0

Views

Author

Seiichi Manyama, May 05 2023

Keywords

Examples

			Square array begins:
     1,    1,    1,   1,   1,     1, ...
     1,    0,   -1,  -2,  -3,    -4, ...
     4,    3,    4,   7,  12,    19, ...
    27,   17,    7,  -9, -37,   -83, ...
   256,  169,  120, 121, 208,   441, ...
  3125, 2079, 1373, 797,  21, -1525, ...

Links

Crossrefs

Columns k=0..3 give A000312, (-1)^n * A069856(n), A362857, A362858.
Main diagonal gives A290158.
Cf. A362019.

Programs

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

Formula

E.g.f. of column k: exp(-k*x) / (1 + LambertW(-x)).
G.f. of column k: Sum_{j>=0} (j*x)^j / (1 + k*x)^(j+1).

A362337 a(n) = n! * Sum_{k=0..floor(n/2)} (-k)^k / (k! * (n-2*k)!).

Original entry on oeis.org

1, 1, -1, -5, 37, 221, -2549, -21041, 342665, 3604537, -75816809, -970017949, 25012223149, 377031935125, -11513789879773, -199833956857289, 7052339905578001, 138505710577529969, -5546345926322582225, -121599560980889072693, 5447342134797972438581
Offset: 0

Views

Author

Seiichi Manyama, Apr 17 2023

Keywords

Links

Crossrefs

Cf. A069856, A362338, A362339.
Cf. A362347.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(x^2))))

Formula

E.g.f.: exp(x) / (1 + LambertW(x^2)).

A362338 a(n) = n! * Sum_{k=0..floor(n/3)} (-k)^k / (k! * (n-3*k)!).

Original entry on oeis.org

1, 1, 1, -5, -23, -59, 1321, 9871, 39985, -1512503, -16027919, -89148509, 4751428441, 65256458125, 461686022617, -31737431328329, -535583971806239, -4599769739165039, 387180506424212065, 7750866424109754187, 78298694889496869961, -7798395141074580424619
Offset: 0

Views

Author

Seiichi Manyama, Apr 17 2023

Keywords

Links

Crossrefs

Cf. A069856, A362337, A362339.
Cf. A362348.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(x^3))))

Formula

E.g.f.: exp(x) / (1 + LambertW(x^3)).

A362339 a(n) = n! * Sum_{k=0..floor(n/4)} (-k)^k / (k! * (n-4*k)!).

Original entry on oeis.org

1, 1, 1, 1, -23, -119, -359, -839, 78961, 722737, 3623761, 13297681, -2115602279, -27917827079, -195909017303, -980236890359, 219254440161121, 3780662914771681, 34105981790126881, 216149350680413857, -62275804867272039479, -1325952502191492278039
Offset: 0

Views

Author

Seiichi Manyama, Apr 17 2023

Keywords

Links

Crossrefs

Cf. A069856, A362337, A362338.
Cf. A362349.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(x^4))))

Formula

E.g.f.: exp(x) / (1 + LambertW(x^4)).

A362340 a(n) = n! * Sum_{k=0..floor(n/2)} (-k/2)^k / (k! * (n-2*k)!).

Original entry on oeis.org

1, 1, 0, -2, 7, 51, -239, -2435, 16353, 209377, -1826099, -28232379, 303020125, 5494172893, -70032035163, -1457369472299, 21512472563281, 505400696581905, -8478758871011807, -221971772323923263, 4171251104170567101, 120416449897739144941
Offset: 0

Views

Author

Seiichi Manyama, Apr 17 2023

Keywords

Links

Crossrefs

Cf. A069856, A362341, A362342.
Cf. A362276.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(x^2/2))))

Formula

E.g.f.: exp(x) / (1 + LambertW(x^2/2)).

A362341 a(n) = n! * Sum_{k=0..floor(n/3)} (-k/6)^k / (k! * (n-3*k)!).

Original entry on oeis.org

1, 1, 1, 0, -3, -9, 21, 246, 1065, -4283, -67319, -397484, 2315941, 45914155, 343743037, -2623221054, -62980998639, -571382718039, 5391435590545, 152175023203432, 1622112809355661, -18232162910685569, -591788241447761819, -7247966654986009490
Offset: 0

Views

Author

Seiichi Manyama, Apr 17 2023

Keywords

Links

Crossrefs

Cf. A069856, A362340, A362342.
Cf. A351929, A362303, A362348.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(x^3/6))))

Formula

E.g.f.: exp(x) / (1 + LambertW(x^3/6)).

A362342 a(n) = n! * Sum_{k=0..floor(n/4)} (-k/24)^k / (k! * (n-4*k)!).

Original entry on oeis.org

1, 1, 1, 1, 0, -4, -14, -34, 71, 1135, 6091, 22771, -87119, -1847559, -13769755, -70046339, 390688481, 10473961121, 100030347361, 643972996705, -4717305354419, -153449916040259, -1787926183752939, -13926752488607419, 126329848106764765
Offset: 0

Views

Author

Seiichi Manyama, Apr 17 2023

Keywords

Links

Crossrefs

Cf. A069856, A362340, A362341.
Cf. A351930, A362345, A362349.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(x^4/24))))

Formula

E.g.f.: exp(x) / (1 + LambertW(x^4/24)).

A065980 Inverse binomial transform of [1^1,2^2,3^3,...], shifted right by one index.

Original entry on oeis.org

1, 3, 20, 186, 2248, 33340, 585744, 11891236, 273854368, 7053523236, 200894140120, 6268924259884, 212691682554960, 7795165961244532, 306908654169113416, 12918649608270463740, 578931362074039774144
Offset: 1

Views

Author

Robert A. Stump (bee_ess107(AT)yahoo.com), Dec 09 2001

Keywords

Comments

{0, a(n),n=1,...} = inverse binomial transform of {A001923(m), m=0,...} [From Tilman Neumann, Dec 17 2008]

Links

Crossrefs

Cf. A001923, A069856.

Programs

  • Mathematica
    CoefficientList[Series[-E^(-x)*LambertW[-x]/(1+LambertW[-x])^3/x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 17 2014 *)
  • PARI
    a(n)=if(n<1,0,(n-1)!*polcoeff(exp(-x+O(x^n))*sum(k=0,n-1,(k+1)^(k+1)*x^k/k!),n-1))

Formula

O.g.f.: Sum_{n>0} (n*x/(1+x))^n. E.g.f.: int(-exp(-x)*LambertW(-x)/(1+LambertW(-x))^3/x, x). - Vladeta Jovovic, Apr 12 2003
a(n) ~ n^n * exp(-exp(-1)). - Vaclav Kotesovec, Feb 17 2014
Previous Showing 11-20 of 20 results.

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