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 1-10 of 25 results. Next

A193363 O.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^n * x^n * A((n+1)*x)^n/n! * exp(-(n+1)*x*A((n+1)*x)).

Original entry on oeis.org

1, 1, 4, 41, 871, 36137, 2885457, 443469511, 131707909982, 75945551138638, 85425571722359386, 188277619627892581987, 816318863956958720950775, 6986374103851011507327849798, 118360360643974268213872443877649, 3978536338453184605328853807076468581
Offset: 0

Views

Author

Paul D. Hanna, Jan 09 2013

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} (n+1)^n * x^n * G(x)^n/n! * exp(-(n+1)*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 4*x^2 + 41*x^3 + 871*x^4 + 36137*x^5 + 2885457*x^6 +...
where
A(x) = exp(-x*A(x)) + 2*x*A(2*x)*exp(-2*x*A(2*x)) + 3^2*x^2*A(3*x)^2/2!*exp(-3*x*A(3*x)) + 4^3*x^3*A(4*x)^3/3!*exp(-4*x*A(4*x)) + 5^4*x^4*A(5*x)^4/4!*exp(-5*x*A(5*x)) + 6^5*x^5*A(6*x)^5/5!*exp(-6*x*A(6*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A209276, A209277.
Cf. A221409, A221410, A221411, A221412, A221413.

Programs

  • Mathematica
    A[_] = 0; m = 16;
Do[A[x_] = Exp[-x A[x]] + Sum[(n+1)^n x^n A[(n+1)x]^n/n! Exp[-(n+1) x A[(n+1)x]], {n, 1, m}] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 29 2019 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+1)^k*x^k*subst(A, x, (k+1)*x)^k/k!*exp(-(k+1)*x*subst(A, x, (k+1)*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

A218681 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n^2*x)^n/n! * exp(-n*x*A(n^2*x)).

Original entry on oeis.org

1, 1, 2, 17, 248, 8044, 499033, 62625238, 15947986557, 8220983161264, 8675909809528468, 18709697284980554577, 82551047593942653184220, 747564468621251440782891798, 13885138852461763218258064204207, 529723356811556257370919794910913765
Offset: 0

Views

Author

Paul D. Hanna, Nov 06 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 2*x^2 + 17*x^3 + 248*x^4 + 8044*x^5 + 499033*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2^2*x)^2/2!*exp(-2*x*A(2^2*x)) + 3^3*x^3*A(3^2*x)^3/3!*exp(-3*x*A(3^2*x)) + 4^4*x^4*A(4^2*x)^4/4!*exp(-4*x*A(4^2*x)) + 5^5*x^5*A(5^2*x)^5/5!*exp(-5*x*A(5^2*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A219342, A185029.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A,x,k^2*x)^k/k!*exp(-k*x*subst(A,x,k^2*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A221409 O.g.f. satisfies: A(x) = Sum_{n>=0} (n-1)^n * x^n * A(n*x)^n/n! * exp(-(n-1)*A(n*x)).

Original entry on oeis.org

1, 1, 1, 3, 16, 117, 1185, 16856, 334597, 9263497, 360493767, 19836684505, 1547142671748, 171456480498151, 27060184630906514, 6089195353964497464, 1955550547239382775017, 897232469707513867626376, 588505259787507511336381953, 552133036731399028180043225074
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2013

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} (n-1)^n * x^n * G(x)^n/n! * exp(-(n-1)*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + x^2 + 3*x^3 + 16*x^4 + 117*x^5 + 1185*x^6 +...
where
A(x) = exp(x) + 0*x*A(x)*exp(-0*x*A(x)) + 1^2*x^2*A(2*x)^2/2!*exp(-1*x*A(2*x)) + 2^3*x^3*A(3*x)^3/3!*exp(-2*x*A(3*x)) + 3^4*x^4*A(4*x)^4/4!*exp(-3*x*A(4*x)) + 4^5*x^5*A(5*x)^5/5!*exp(-4*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A193363, A221410, A221411, A221412, A221413.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k-1)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k-1)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

A221410 O.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^n * x^n * A(n*x)^n/n! * exp(-(n+1)*x*A(n*x)).

Original entry on oeis.org

1, 1, 3, 17, 160, 2209, 44081, 1247278, 50003383, 2843143785, 229717311597, 26423288336013, 4331881870569310, 1013060852125392519, 338180578288458076194, 161225876602752196310870, 109821236456762132613619651, 106923122485613725232770276036
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2013

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} (n+1)^n * x^n * G(x)^n/n! * exp(-(n+1)*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 160*x^4 + 2209*x^5 + 44081*x^6 +...
where
A(x) = exp(-x) + 2*x*A(x)*exp(-2*x*A(x)) + 3^2*x^2*A(2*x)^2/2!*exp(-3*x*A(2*x)) + 4^3*x^3*A(3*x)^3/3!*exp(-4*x*A(3*x)) + 5^4*x^4*A(4*x)^4/4!*exp(-5*x*A(4*x)) + 6^5*x^5*A(5*x)^5/5!*exp(-6*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A193363, A221409, A221411, A221412, A221413.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+1)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+1)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

A221411 O.g.f. satisfies: A(x) = Sum_{n>=0} (n+2)^n * x^n * A(n*x)^n/n! * exp(-(n+2)*x*A(n*x)).

Original entry on oeis.org

1, 1, 4, 27, 325, 5553, 140103, 4993445, 253780210, 18321882898, 1882677322704, 275715048156637, 57570654555092091, 17152947168669102772, 7295365645092117304955, 4430848642167010373612127, 3844378527942068170940925685, 4766454891141869269974497298382
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2013

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} (n+2)^n * x^n * G(x)^n/n! * exp(-(n+2)*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 4*x^2 + 27*x^3 + 325*x^4 + 5553*x^5 + 140103*x^6 +...
where
A(x) = exp(-2*x) + 3*x*A(x)*exp(-3*x*A(x)) + 4^2*x^2*A(2*x)^2/2!*exp(-4*x*A(2*x)) + 5^3*x^3*A(3*x)^3/3!*exp(-5*x*A(3*x)) + 6^4*x^4*A(4*x)^4/4!*exp(-6*x*A(4*x)) + 7^5*x^5*A(5*x)^5/5!*exp(-7*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A193363, A221409, A221410, A221412, A221413.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+2)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+2)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

A221412 O.g.f. satisfies: A(x) = Sum_{n>=0} (n+3)^n * x^n * A(n*x)^n/n! * exp(-(n+3)*x*A(n*x)).

Original entry on oeis.org

1, 1, 5, 39, 576, 11693, 358649, 15411564, 951579001, 83392989241, 10419431480203, 1856210104355977, 471536928543684056, 170959559745467848287, 88469465053214549982042, 65371115770077488407503980, 68993903807593031325051425205, 104033290140443202579946504758992
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2013

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} (n+3)^n * x^n * G(x)^n/n! * exp(-(n+3)*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 5*x^2 + 39*x^3 + 576*x^4 + 11693*x^5 + 358649*x^6 +...
where
A(x) = exp(-3*x) + 4*x*A(x)*exp(-4*x*A(x)) + 5^2*x^2*A(2*x)^2/2!*exp(-5*x*A(2*x)) + 6^3*x^3*A(3*x)^3/3!*exp(-6*x*A(3*x)) + 7^4*x^4*A(4*x)^4/4!*exp(-7*x*A(4*x)) + 8^5*x^5*A(5*x)^5/5!*exp(-8*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A193363, A221409, A221410, A221411, A221413.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+3)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+3)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

A221413 O.g.f. satisfies: A(x) = Sum_{n>=0} (n+4)^n * x^n * A(n*x)^n/n! * exp(-(n+4)*x*A(n*x)).

Original entry on oeis.org

1, 1, 6, 53, 931, 21847, 791525, 39781921, 2896348222, 298603689072, 43979877929712, 9234821696038425, 2765498896234870783, 1182132922860352133076, 721128788569371093881079, 628104461090874688307332589, 781298529318782688558174387547
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2013

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 6*x^2 + 53*x^3 + 931*x^4 + 21847*x^5 + 791525*x^6 +...
where
A(x) = exp(-4*x) + 5*x*A(x)*exp(-5*x*A(x)) + 6^2*x^2*A(2*x)^2/2!*exp(-6*x*A(2*x)) + 7^3*x^3*A(3*x)^3/3!*exp(-7*x*A(3*x)) + 8^4*x^4*A(4*x)^4/4!*exp(-8*x*A(4*x)) + 9^5*x^5*A(5*x)^5/5!*exp(-9*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A193363, A221409, A221410, A221411, A221412.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+4)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+4)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

A218673 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(2*n)/n! * exp(-n*x*A(n*x)^2).

Original entry on oeis.org

1, 1, 3, 20, 209, 3173, 67292, 1970761, 79764057, 4490097388, 354111363537, 39360693851404, 6193012446752244, 1383433132321835172, 439684769985895688173, 199116777197880585373014, 128631139424158036273736167, 118640007280899188486618513612
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 3*x^2 + 20*x^3 + 209*x^4 + 3173*x^5 + 67292*x^6 +...
where
A(x) = 1 + x*A(x)^2*exp(-x*A(x)^2) + 2^2*x^2*A(2*x)^4/2!*exp(-2*x*A(2*x)^2) + 3^3*x^3*A(3*x)^6/3!*exp(-3*x*A(3*x)^2) + 4^4*x^4*A(4*x)^8/4!*exp(-4*x*A(4*x)^2) + 5^5*x^5*A(5*x)^10/5!*exp(-5*x*A(5*x)^2) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A218674, A218675, A218676.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^2,x,k*x)^k/k!*exp(-k*x*subst(A^2,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A218674 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(3*n)/n! * exp(-n*x*A(n*x)^3).

Original entry on oeis.org

1, 1, 4, 34, 455, 8710, 230077, 8285224, 407456797, 27587687551, 2596034329278, 342275007167359, 63606742005546232, 16730509857101195808, 6246818082857455197662, 3317816101992338134691233, 2510420393373091580780786808, 2709148467943025007607468405672
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 455*x^4 + 8710*x^5 + 230077*x^6 +...
where
A(x) = 1 + x*A(x)^3*exp(-x*A(x)^3) + 2^2*x^2*A(2*x)^6/2!*exp(-2*x*A(2*x)^3) + 3^3*x^3*A(3*x)^9/3!*exp(-3*x*A(3*x)^3) + 4^4*x^4*A(4*x)^12/4!*exp(-4*x*A(4*x)^3) + 5^5*x^5*A(5*x)^15/5!*exp(-5*x*A(5*x)^3) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A218673, A218675, A218676.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^3,x,k*x)^k/k!*exp(-k*x*subst(A^3,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A218675 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(4*n)/n! * exp(-n*x*A(n*x)^4).

Original entry on oeis.org

1, 1, 5, 51, 817, 18562, 576687, 24203258, 1375038677, 106708683355, 11435867474152, 1708844338589752, 358640659116617571, 106261016900832212139, 44607231638918264608274, 26598477338494285370797703, 22569718290467849884279856477
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 5*x^2 + 51*x^3 + 817*x^4 + 18562*x^5 + 576687*x^6 +...
where
A(x) = 1 + x*A(x)^4*exp(-x*A(x)^4) + 2^2*x^2*A(2*x)^8/2!*exp(-2*x*A(2*x)^4) + 3^3*x^3*A(3*x)^12/3!*exp(-3*x*A(3*x)^4) + 4^4*x^4*A(4*x)^16/4!*exp(-4*x*A(4*x)^4) + 5^5*x^5*A(5*x)^20/5!*exp(-5*x*A(5*x)^4) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A218673, A218674, A218676.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^4,x,k*x)^k/k!*exp(-k*x*subst(A^4,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
Showing 1-10 of 25 results. Next

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