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 1-5 of 5 results.

A273954 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * exp(n*x) * A(x)^n.

Original entry on oeis.org

1, 1, 5, 37, 393, 5481, 95053, 1975821, 47939601, 1330923601, 41629292181, 1448989481589, 55561575788953, 2327512861252281, 105767732851318749, 5182512561142513501, 272391086209524010017, 15287595381259195453089, 912525533175190887597349, 57726267762799335649572549
Offset: 0

Views

Author

Paul D. Hanna, Jun 14 2016

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 393*x^4/4! + 5481*x^5/5! + 95053*x^6/6! + 1975821*x^7/7! + 47939601*x^8/8! + 1330923601*x^9/9! + 41629292181*x^10/10! + 1448989481589*x^11/11! + 55561575788953*x^12/12! +...
such that
A(x) = 1 + x*exp(x)*A(x) + x^2/2!*exp(2*x)*A(x)^2 + x^3/3!*exp(3*x)*A(x)^3 + x^4/4!*exp(4*x)*A(x)^4 + x^5/5!*exp(5*x)*A(x)^5 + x^6/6!*exp(6*x)*A(x)^6 +...
The logarithm of A(x) begins:
log(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 47232*x^6/6! + 942592*x^7/7! + 22171648*x^8/8! + 600698880*x^9/9! + 18422374400*x^10/10! +...+ A216857(n)*x^n/n! +...
which equals -LambertW(-x*exp(x)).

Links

Crossrefs

Cf. A273953, A216857, A357247, A360176 (column 1 unsigned).
Cf. A161628, A244119.

Programs

  • Maple
    A273954 := n -> add(binomial(n, j) * j^(n - j) * (j + 1)^(j - 1), j = 0..n):
seq(A273954(n), n = 0..24); # Peter Luschny, Jan 29 2023
  • Mathematica
    CoefficientList[Series[-LambertW[-x*E^x] / (x*E^x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 23 2016 *)
  • PARI
    {a(n) = my(A=1+x); for(i=1,n, A = sum(m=0,n,x^m/m!*exp(m*x +x*O(x^n))*A^m) ); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    x='x+O('x^50); Vec(serlaplace(-lambertw(-x*exp(x))/(x*exp(x)))) \\ G. C. Greubel, Nov 16 2017
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x*exp(x))^k/k!))) \\ Seiichi Manyama, Feb 08 2023

Formula

E.g.f.: -LambertW(-x*exp(x)) / (x*exp(x)). [corrected by Vaclav Kotesovec, Jun 23 2016]
E.g.f.: exp( L(x) ) where L(x) = -LambertW(-x*exp(x)) is the e.g.f. of A216857.
a(n) ~ sqrt(1+LambertW(exp(-1))) * n^(n-1) / (exp(n-1) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Jun 23 2016
E.g.f.: A(x) = exp(x*exp(x)*A(x)). - Alexander Burstein, Aug 11 2018
From Peter Luschny, Jan 29 2023: (Start)
a(n) = Sum_{j=0..n} binomial(n, j) * j^(n - j) * (j + 1)^(j - 1).
a(n) = Sum_{k=0..n} (-1)^k*A161628(n, k).
a(n) = Sum_{k=0..n} (-1)^(n-k)*A244119(n, k). (End)

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

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

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

A372279 E.g.f. A(x) satisfies A(x) = exp( x * ( exp(x) * A(x) )^(5/2) ).

Original entry on oeis.org

1, 1, 11, 181, 4461, 148101, 6202651, 314158461, 18682884681, 1276509416761, 98552772971451, 8485633118339301, 806247602665104661, 83796784405535693181, 9457590223483413296811, 1151924494605809502276301, 150602291336042725831941201
Offset: 0

Views

Author

Seiichi Manyama, Apr 25 2024

Keywords

Links

Crossrefs

Cf. A273953, A273954, A360544, A360547.
Cf. A372278.

Programs

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

Formula

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

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