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 1-8 of 8 results.

A360547 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^2 ).

Original entry on oeis.org

1, 1, 9, 121, 2417, 64721, 2180665, 88719625, 4233968737, 231991022881, 14356691152361, 990506937621785, 75390334060230865, 6275675303410022641, 567191776288882702105, 55313848534122299876521, 5789703106014903009828545, 647414950001156861671249985
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2023

Keywords

Links

Crossrefs

Cf. A273953, A273954, A360544.
Cf. A360465, A360473.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x]*A[x])^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*exp(2*x))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(-lambertw(-2*x*exp(2*x))/(2*x*exp(2*x)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(sum(k=0, N, (k+1)^(k-1)*(2*x*exp(2*x))^k/k!))))
  • PARI
    a(n) = sum(k=0, n, (2*k)^(n-k)*(2*k+1)^(k-1)*binomial(n, k));

Formula

E.g.f.: A(x) = exp( (-1/2) * LambertW(-2*x * exp(2*x)) ).
E.g.f.: A(x) = sqrt( -LambertW(-2*x * exp(2*x)) / (2*x * exp(2*x)) ).
E.g.f.: A(x) = sqrt( Sum_{k>=0} (k+1)^(k-1) * (2*x * exp(2*x))^k / k! ).
a(n) = Sum_{k=0..n} (2*k)^(n-k) * (2*k+1)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (exp(n - 1/2) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Feb 17 2023

A360474 E.g.f. satisfies A(x) = exp( x * A(x)^2 * exp(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 7, 94, 1921, 53036, 1849789, 78070462, 3869909537, 220427550712, 14188370562901, 1018570771664546, 80692202644742737, 6992855583524143204, 658076908751441373965, 66833181471569822199886, 7285736943975575120653249, 848589321771735983890457072
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2023

Keywords

Crossrefs

Cf. A162695, A360473.

Programs

  • Mathematica
    Join[{1}, Table[Sum[k^(n-k) * (2*n+1)^(k-1) * Binomial[n,k], {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Feb 17 2023 *)
  • PARI
    a(n) = sum(k=0, n, k^(n-k)*(2*n+1)^(k-1)*binomial(n, k));

Formula

a(n) = Sum_{k=0..n} k^(n-k) * (2*n+1)^(k-1) * binomial(n,k).
a(n) ~ 2^(n - 1/2) * n^(n-1) * s^(2*n + 1) * log(s)^(n + 1/2) / (sqrt(1 + 2*log(s) - 4*log(s)^2) * exp(n) * (1 - 2*log(s))^n), where s = 1.473428520956658037187728756446912746332041803082... is the root of the equation 2*log(s)*(1 + LambertW(log(s))) = 1. - Vaclav Kotesovec, Feb 17 2023

A360481 E.g.f. satisfies A(x) = x * exp(x + 2 * A(x)).

Original entry on oeis.org

0, 1, 6, 63, 1044, 23805, 692118, 24482115, 1020584232, 49000005945, 2662853279850, 161586078510879, 10830019921469532, 794577001293803637, 63339899145968483262, 5451312770064188283195, 503784284643602483767632, 49757423537114340032969073
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2023

Keywords

Links

Crossrefs

Cf. A216857, A360482, A360483, A360484.
Cf. A360473.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x*exp(x))/2)))
  • PARI
    a(n) = sum(k=1, n, 2^(k-1)*k^(n-1)*binomial(n, k));

Formula

E.g.f.: A(x) = -LambertW(-2*x * exp(x))/2.
a(n) = Sum_{k=1..n} 2^(k-1) * k^(n-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1)/2)) * n^(n-1) / (2 * LambertW(exp(-1)/2)^n * exp(n)). - Vaclav Kotesovec, Feb 17 2023

A360544 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^(3/2) ).

Original entry on oeis.org

1, 1, 7, 73, 1117, 22741, 580159, 17826985, 641494249, 26473635865, 1232945359111, 63978649829161, 3660871368065509, 229016870623703917, 15550838554432967647, 1139139301403727884521, 89544381521098908259729, 7518611017848248249471089
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2023

Keywords

Links

Crossrefs

Cf. A273953, A273954, A360547.
Cf. A360473, A360545.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*lambertw(-3/2*x*exp(3*x/2))/3)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-3*x/2*exp(3*x/2))/(3*x/2*exp(3*x/2)))^(2/3)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((sum(k=0, N, (k+1)^(k-1)*(3*x/2*exp(3*x/2))^k/k!))^(2/3)))
  • PARI
    a(n) = sum(k=0, n, (3*k)^(n-k)*(3*k+2)^(k-1)*binomial(n, k))/2^(n-1);

Formula

E.g.f.: A(x) = exp( (-2/3) * LambertW(-3*x/2 * exp(3*x/2)) ).
E.g.f.: A(x) = ( -LambertW(-3*x/2 * exp(3*x/2)) / (3*x/2 * exp(3*x/2)) )^(2/3).
E.g.f.: A(x) = ( Sum_{k>=0} (k+1)^(k-1) * (3*x/2 * exp(3*x/2))^k / k! )^(2/3).
a(n) = (1/2^(n-1)) * Sum_{k=0..n} (3*k)^(n-k) * (3*k+2)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^(n-1) * exp(n - 2/3) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Feb 17 2023

A362656 E.g.f. satisfies A(x) = exp( x * exp(x) * A(x)^3 ).

Original entry on oeis.org

1, 1, 9, 145, 3569, 119041, 5025145, 256991953, 15448193633, 1067634195841, 83414064659561, 7270683884044945, 699503964027087697, 73631519384051331457, 8417768844410686595801, 1038658083084399115865041, 137579671405398060549801665
Offset: 0

Views

Author

Seiichi Manyama, Apr 28 2023

Keywords

Links

Crossrefs

Cf. A273954, A360473, A362671, A362672.
Cf. A052752.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-3*x*exp(x))/3)))

Formula

E.g.f.: exp( -LambertW(-3*x * exp(x))/3 ).
a(n) = Sum_{k=0..n} k^(n-k) * (3*k+1)^(k-1) * binomial(n,k).

A362671 E.g.f. satisfies A(x) = exp( x * exp(x) / A(x)^2 ).

Original entry on oeis.org

1, 1, -1, 10, -111, 1716, -33755, 807738, -22782207, 740204776, -27226430739, 1118416240470, -50750734988063, 2521219487859372, -136098630522431499, 7932551567421395866, -496501182232557828735, 33214032504796887027408
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2023

Keywords

Links

Crossrefs

Cf. A273954, A360473, A362656, A362672.

Programs

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

Formula

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

A362672 E.g.f. satisfies A(x) = exp( x * exp(x) / A(x)^3 ).

Original entry on oeis.org

1, 1, -3, 37, -679, 17161, -553451, 21731053, -1006118863, 53671172113, -3241671266899, 218677223408821, -16296163119155063, 1329568681331536153, -117874745761237043515, 11283758432396431997821, -1159952212029532257663391, 127445385808282289840496673
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2023

Keywords

Links

Crossrefs

Cf. A273954, A360473, A362656, A362671.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*x*exp(x))/3)))

Formula

E.g.f.: exp( LambertW(3*x * exp(x))/3 ).
a(n) = Sum_{k=0..n} k^(n-k) * (-3*k+1)^(k-1) * binomial(n,k).

A380406 E.g.f. satisfies A(x) = exp( 2 * x * exp(x) * A(x)^(1/2) ).

Original entry on oeis.org

1, 2, 12, 104, 1232, 18592, 342208, 7451264, 187631872, 5369721344, 172255038464, 6125052946432, 239195824279552, 10179739052908544, 469024768235192320, 23263095316577681408, 1235978286454556131328, 70040404736026578386944, 4217180561907991530176512
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2025

Keywords

Links

Crossrefs

Cf. A273954, A357247, A380407.
Cf. A273953, A360473.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A273954.
E.g.f.: A(x) = exp( -2*LambertW(-x * exp(x)) ).
a(n) = 2 * Sum_{k=0..n} k^(n-k) * (k+2)^(k-1) * binomial(n,k).
a(n) ~ 2 * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n-2) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Aug 05 2025
Showing 1-8 of 8 results.

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