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

A349716 E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^5)/2 ).

Original entry on oeis.org

1, 1, 6, 91, 2156, 69926, 2884576, 144555356, 8529135216, 579220982056, 44503081624976, 3816776859516776, 361462121953291456, 37464997600663289216, 4218485281787859411456, 512762346462142021355776, 66919363061333997572830976, 9332997074366800051673277056
Offset: 0

Views

Author

Seiichi Manyama, Nov 26 2021

Keywords

Links

Crossrefs

Cf. A007889, A202617, A349714, A349715, A349719, A349720, A349721.

Programs

  • Mathematica
    a[n_] := (1/2^n) * Sum[(5*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, (5*k+1)^(n-1)*binomial(n, k))/2^n;
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-5*x/2*exp(5*x/2))/(5*x/2))^(1/5)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (5*k+1)^(k-1)*x^k/(2-(5*k+1)*x)^(k+1)))

Formula

a(n) = (1/2^n) * Sum_{k=0..n} (5*k+1)^(n-1) * binomial(n,k).
E.g.f.: ( -LambertW( -5*x/2 * exp(5*x/2) )/(5*x/2) )^(1/5).
G.f.: 2 * Sum_{k>=0} (5*k+1)^(k-1) * x^k/(2 - (5*k+1)*x)^(k+1).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 5^(n-1) * n^(n-1) / (LambertW(exp(-1))^(n + 1/5) * 2^n * exp(n)). - Vaclav Kotesovec, Nov 26 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

A362735 E.g.f. satisfies A(x) = exp(x + x / A(x)^2).

Original entry on oeis.org

1, 2, -4, 56, -1008, 25632, -833600, 33067904, -1548418816, 83597525504, -5112566055936, 349330707068928, -26374805535322112, 2180554321981349888, -195926186031705505792, 19010400989418574020608, -1980997069982960384409600, 220651645970702249702326272
Offset: 0

Views

Author

Seiichi Manyama, May 01 2023

Keywords

Links

Crossrefs

Cf. A349562, A362693, A362694, A362734.
Cf. A349720.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x+lambertw(2*x*exp(-2*x))/2)))

Formula

E.g.f.: sqrt( 2*x / LambertW(2*x*exp(-2*x)) ) = exp( x + LambertW(2*x*exp(-2*x))/2 ).
a(n) = Sum_{k=0..n} (-2*k+1)^(n-1) * binomial(n,k) = 2^n * A349720(n).
Showing 1-6 of 6 results.

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

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