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 1-6 of 6 results.

A349719 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x))/2 ).

Original entry on oeis.org

1, 1, 0, 1, -4, 26, -212, 2108, -24720, 334072, -5112544, 87396728, -1650607040, 34132685120, -767025716736, 18612106195456, -485013257865472, 13509071081429888, -400505695457942528, 12592502771190979712, -418524228123134068224
Offset: 0

Views

Author

Seiichi Manyama, Nov 27 2021

Keywords

Links

Crossrefs

Cf. A007889, A202617, A283828, A349714, A349715, A349716, A349720, A349721.

Programs

  • Mathematica
    a[n_] := (1/2^n) * Sum[If[k == n == 1, 1, (-k + 1)^(n - 1)] * Binomial[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    a(n) = sum(k=0, n, (-k+1)^(n-1)*binomial(n, k))/2^n;
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace((x/2)/lambertw(x/2*exp(-x/2))))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(2*sum(k=0, N, (-k+1)^(k-1)*x^k/(2-(-k+1)*x)^(k+1)))

Formula

a(n) = (1/2^n) * Sum_{k=0..n} (-k+1)^(n-1) * binomial(n,k).
E.g.f.: (x/2)/LambertW( x/2 * exp(-x/2) ).
G.f.: 2 * Sum_{k>=0} (-k+1)^(k-1) * x^k/(2 - (-k+1)*x)^(k+1).
a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (2^n * exp(n) * LambertW(exp(-1))^(n-1)). - Vaclav Kotesovec, Dec 05 2021

A349714 E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^3)/2 ).

Original entry on oeis.org

1, 1, 4, 37, 532, 10426, 259300, 7823908, 277713904, 11339452792, 523621438336, 26982030104536, 1534947906550528, 95550736737542464, 6460746383585984512, 471533064029919744256, 36946948091091750496000, 3093472887944746070621056
Offset: 0

Views

Author

Seiichi Manyama, Nov 26 2021

Keywords

Links

Crossrefs

Cf. A007889, A202617, A349715, A349716, A349719, A349720, A349721.

Programs

  • Mathematica
    a[n_] := (1/2^n) * Sum[(3*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    a(n) = sum(k=0, n, (3*k+1)^(n-1)*binomial(n, k))/2^n;
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-3*x/2*exp(3*x/2))/(3*x/2))^(1/3)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (3*k+1)^(k-1)*x^k/(2-(3*k+1)*x)^(k+1)))

Formula

a(n) = (1/2^n) * Sum_{k=0..n} (3*k+1)^(n-1) * binomial(n,k).
E.g.f.: ( -LambertW( -3*x/2 * exp(3*x/2) )/(3*x/2) )^(1/3).
G.f.: 2 * Sum_{k>=0} (3*k+1)^(k-1) * x^k/(2 - (3*k+1)*x)^(k+1).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (LambertW(exp(-1))^(n + 1/3) * 2^n * exp(n)). - Vaclav Kotesovec, Nov 26 2021

A349720 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^2)/2 ).

Original entry on oeis.org

1, 1, -1, 7, -63, 801, -13025, 258343, -6048511, 163276417, -4992740289, 170571634311, -6439161507647, 266180947507489, -11958385377911713, 580151397382158631, -30227616424300542975, 1683438461080186841601, -99796591057813372007297
Offset: 0

Views

Author

Seiichi Manyama, Nov 27 2021

Keywords

Links

Crossrefs

Cf. A007889, A202617, A349714, A349715, A349716, A349719, A349721.

Programs

  • Mathematica
    a[n_] := (1/2^n) * Sum[(-2*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    a(n) = sum(k=0, n, (-2*k+1)^(n-1)*binomial(n, k))/2^n;
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((x/lambertw(x*exp(-x)))^(1/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (-2*k+1)^(k-1)*x^k/(2-(-2*k+1)*x)^(k+1)))

Formula

a(n) = (1/2^n) * Sum_{k=0..n} (-2*k+1)^(n-1) * binomial(n,k).
E.g.f.: ( x/LambertW( x * exp(-x) ) )^(1/2).
G.f.: 2 * Sum_{k>=0} (-2*k+1)^(k-1) * x^k/(2 - (-2*k+1)*x)^(k+1).
a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (2 * exp(n) * LambertW(exp(-1))^(n - 1/2)). - Vaclav Kotesovec, Dec 05 2021

A349715 E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^4)/2 ).

Original entry on oeis.org

1, 1, 5, 61, 1161, 30201, 998413, 40077493, 1893550865, 102951388657, 6331847746581, 434653328279853, 32944254978940825, 2732662648183661545, 246228744062320481309, 23949858491053731087781, 2501088964314980938821153, 279111248034686114681365473
Offset: 0

Views

Author

Seiichi Manyama, Nov 26 2021

Keywords

Links

Crossrefs

Cf. A007889, A202617, A349714, A349716, A349719, A349720, A349721.

Programs

  • Mathematica
    a[n_] := (1/2^n) * Sum[(4*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    a(n) = sum(k=0, n, (4*k+1)^(n-1)*binomial(n, k))/2^n;
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-2*x*exp(2*x))/(2*x))^(1/4)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (4*k+1)^(k-1)*x^k/(2-(4*k+1)*x)^(k+1)))

Formula

a(n) = (1/2^n) * Sum_{k=0..n} (4*k+1)^(n-1) * binomial(n,k).
E.g.f.: ( -LambertW( -2*x * exp(2*x) )/(2*x) )^(1/4).
G.f.: 2 * Sum_{k>=0} (4*k+1)^(k-1) * x^k/(2 - (4*k+1)*x)^(k+1).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-2) * n^(n-1) / (LambertW(exp(-1))^(n + 1/4) * exp(n)). - Vaclav Kotesovec, Nov 26 2021

A349721 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^3)/2 ).

Original entry on oeis.org

1, 1, -2, 19, -260, 4966, -121328, 3613996, -127035920, 5147600680, -236245559984, 12112405259560, -686148484748480, 42560312499982720, -2868921992458611200, 208828244778853125376, -16324500711130356582656, 1363986660232205656646272
Offset: 0

Views

Author

Seiichi Manyama, Nov 27 2021

Keywords

Links

Crossrefs

Cf. A007889, A202617, A349714, A349715, A349716, A349719, A349720.

Programs

  • Mathematica
    a[n_] := (1/2^n) * Sum[(-3*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    a(n) = sum(k=0, n, (-3*k+1)^(n-1)*binomial(n, k))/2^n;
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(((3*x/2)/lambertw(3*x/2*exp(-3*x/2)))^(1/3)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (-3*k+1)^(k-1)*x^k/(2-(-3*k+1)*x)^(k+1)))

Formula

a(n) = (1/2^n) * Sum_{k=0..n} (-3*k+1)^(n-1) * binomial(n,k).
E.g.f.: ( (3*x/2)/LambertW( 3*x/2 * exp(-3*x/2) ) )^(1/3).
G.f.: 2 * Sum_{k>=0} (-3*k+1)^(k-1) * x^k/(2 - (-3*k+1)*x)^(k+1).
a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^n * exp(n) * LambertW(exp(-1))^(n - 1/3)). - Vaclav Kotesovec, Dec 05 2021

A372278 E.g.f. A(x) satisfies A(x) = exp( x * (1 + A(x)^(5/2)) ).

Original entry on oeis.org

1, 2, 14, 218, 5256, 172332, 7161964, 360849848, 21378442976, 1456505344592, 112197636802224, 9643110922761648, 914870017865191936, 94969006015521439232, 10707303771557931935744, 1302965738334245437242368, 170216425515761065556430336
Offset: 0

Views

Author

Seiichi Manyama, Apr 25 2024

Keywords

Links

Crossrefs

Cf. A349562, A362694, A362734, A372236, A372235.
Cf. A349716, A372279.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-2/5*lambertw(-5*x/2*exp(5*x/2)))))
  • PARI
    a(n, r=1, t=0, u=5/2) = r*sum(k=0, n, (t*n+u*k+r)^(n-1)*binomial(n, k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (5*k/2+1)^(k-1)*x^k/(1-(5*k/2+1)*x)^(k+1)))

Formula

E.g.f.: A(x) = exp( x - 2/5 * LambertW(-5*x/2 * exp(5*x/2)) ).
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + A(x)^(u/r)) ), then a(n) = r * Sum_{k=0..n} (t*n+u*k+r)^(n-1) * binomial(n,k).
G.f.: Sum_{k>=0} (5*k/2+1)^(k-1) * x^k/(1 - (5*k/2+1)*x)^(k+1).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 5^(n-1) * n^(n-1) / (2^(n-1) * LambertW(exp(-1))^(n + 2/5) * exp(n)). - Vaclav Kotesovec, May 06 2024
Showing 1-6 of 6 results.

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