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.

A218672 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n*x)^n/n! * exp(-n*x*A(n*x)).

Original entry on oeis.org

1, 1, 2, 9, 63, 659, 9833, 206961, 6133990, 256650268, 15213478000, 1281205909177, 153588353066135, 26245044813624300, 6399076697684238375, 2227912079081482302977, 1108302173165578509079527, 788171767077184315422131588, 801638519723021288783092512047
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare to the LambertW identities:
(1) Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).
(2) Sum_{n>=0} n^n * x^n * C(x)^n/n! * exp(-n*x*C(x)) = C(x), where C(x) = 1 + x*C(x)^2 is the o.g.f. of the Catalan numbers (A000108).

Examples

			O.g.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 63*x^4 + 659*x^5 + 9833*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2*x)^2/2!*exp(-2*x*A(2*x)) + 3^3*x^3*A(3*x)^3/3!*exp(-3*x*A(3*x)) + 4^4*x^4*A(4*x)^4/4!*exp(-4*x*A(4*x)) + 5^5*x^5*A(5*x)^5/5!*exp(-5*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Links

Crossrefs

Cf. A218673, A218674, A218675, A218676, A218681.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.
Cf. A193363, A221409, A221410, A221411, A221412, A221413.
Cf. A209276, A209277.

Programs

  • Mathematica
    a[n_] := Module[{A}, A[x_] = 1 + x; For[i = 1, i <= n, i++, A[x_] = Sum[If[k == 0, 1, k^k] x^k A[k x]^k/k! Exp[-k x A[k x] + x O[x]^i] // Normal, {k, 0, n}]]; Coefficient[ A[x], x, n]];
a /@ Range[0, 18] (* Jean-François Alcover, Sep 29 2019 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A,x,k*x)^k/k!*exp(-k*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A219184 O.g.f. satisfies: A(x) = Sum_{n>=0} n^(2*n) * x^n * A(x)^n / n! * exp(-n^2*x*A(x)).

Original entry on oeis.org

1, 1, 8, 112, 2202, 55641, 1724050, 63550446, 2725133134, 133546286188, 7370574862110, 452601918694564, 30610161317492690, 2260721225822606054, 181023122013996360316, 15619416644091171417138, 1444615406376578862379054, 142565035949775130517868740
Offset: 0

Views

Author

Paul D. Hanna, Nov 13 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 + 8*x^2 + 112*x^3 + 2202*x^4 + 55641*x^5 + 1724050*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^4*x^2*A(x)^2/2!*exp(-4*x*A(x)) + 3^6*x^3*A(x)^3/3!*exp(-9*x*A(x)) + 4^8*x^4*A(x)^4/4!*exp(-16*x*A(x)) + 5^10*x^5*A(x)^5/5!*exp(-25*x*A(x)) +...
simplifies to a power series in x with integer coefficients.
O.g.f. A(x) satisfies A(x) = G(x*A(x)) where G(x) = A(x/G(x)) begins:
G(x) = 1 + x + 7*x^2 + 90*x^3 + 1701*x^4 + 42525*x^5 + 1323652*x^6 +...+ Stirling2(2*n,n)*x^n +...
so that A(x) = (1/x)*Series_Reversion(x/G(x)).

Crossrefs

Cf. A007820, A219228, A217900, A218681, A218672.

Programs

  • PARI
    {a(n)=local(A=1);for(i=1,n,A=sum(m=0, n, (m^2*x*A)^m/m!*exp(-m^2*x*A+x*O(x^n))));polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

Formula

O.g.f. satisfies: A(x) = Sum_{n>=0} Stirling2(2*n,n) * x^n * A(x)^n.

A219342 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n^3*x)^n/n! * exp(-n*x*A(n^3*x)).

Original entry on oeis.org

1, 1, 2, 33, 939, 101175, 26230876, 21032800086, 48319626581926, 319633065306440005, 6299181667747767151873, 359980854813102654362716667, 60552379844778585329083453881153, 30125614945616982039421647789900799744, 43971297878008421196972637327280065832735828
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 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 + 33*x^3 + 939*x^4 + 101175*x^5 + 26230876*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2^3*x)^2/2!*exp(-2*x*A(2^3*x)) + 3^3*x^3*A(3^3*x)^3/3!*exp(-3*x*A(3^3*x)) + 4^4*x^4*A(4^3*x)^4/4!*exp(-4*x*A(4^3*x)) + 5^5*x^5*A(5^3*x)^5/5!*exp(-5*x*A(5^3*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A218681, A185029, A219343, A217900.

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^3*x)^k/k!*exp(-k*x*subst(A,x,k^3*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A185029 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n^4*x)^n/n! * exp(-n*x*A(n^4*x)).

Original entry on oeis.org

1, 1, 2, 65, 3524, 1364432, 1445333132, 7913299718555, 162327934705456532, 14083866155101076361024, 5251111824344114834186373747, 7956883819596423111541696080219295, 51760975171209084256721290749117849746987, 1424616119143714906580708999710589586791029920856
Offset: 0

Views

Author

Paul D. Hanna, Dec 24 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 + 65*x^3 + 3524*x^4 + 1364432*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2^4*x)^2/2!*exp(-2*x*A(2^4*x)) + 3^3*x^3*A(3^4*x)^3/3!*exp(-3*x*A(3^4*x)) + 4^4*x^4*A(4^4*x)^4/4!*exp(-4*x*A(4^4*x)) + 5^5*x^5*A(5^4*x)^5/5!*exp(-5*x*A(5^4*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A218681, A219342.

A219228 O.g.f. satisfies: A(x) = Sum_{n>=0} A(x)^n * (n^3*x)^n/n! * exp(-n^3*x*A(x)).

Original entry on oeis.org

1, 1, 32, 3119, 625710, 214333471, 112105268136, 83149960883200, 83014425998481126, 107334569041127441462, 174471878478682785998864, 348242875992753988109552778, 837327855535084109106340786272, 2387108242583316451939303856237037
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2012

Keywords

Comments

Compare to the o.g.f. of A217913:
Sum_{n>=0} (n^3)^n*exp(-n^3*x)*x^n/n! = Sum_{n>=0} Stirling2(3*n,n)*x^n.

Examples

			O.g.f.: A(x) = 1 + x + 32*x^2 + 3119*x^3 + 625710*x^4 + 214333471*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^6*x^2*A(x)^2/2!*exp(-8*x*A(x)) + 3^9*x^3*A(x)^3/3!*exp(-27*x*A(x)) + 4^12*x^4*A(x)^4/4!*exp(-64*x*A(x)) + 5^15*x^5*A(x)^5/5!*exp(-125*x*A(x)) +...
simplifies to a power series in x with integer coefficients.
G.f. A(x) satisfies A(x) = G(x*A(x)) where G(x) = A(x/G(x)) begins:
G(x) = 1 + x + 31*x^2 + 3025*x^3 + 611501*x^4 + 210766920*x^5 + 110687251039*x^6 +...+ Stirling2(3*n,n)*x^n +...

Crossrefs

Cf. A217913, A219184, A217900, A218681, A218672.

Programs

  • PARI
    {a(n)=local(A=1);for(i=1,n,A=sum(m=0, n, (m^3*x*A)^m/m!*exp(-m^3*x*A+x*O(x^n))));polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

Formula

O.g.f. satisfies: A(x) = Sum_{n>=0} Stirling2(3*n,n) * x^n * A(x)^n.

A219264 O.g.f. satisfies: A(x) = Sum_{n>=0} A(n*x)^n * (n^2*x)^n/n! * exp(-n^2*x*A(n*x)).

Original entry on oeis.org

1, 1, 8, 128, 3259, 120082, 6151625, 433404057, 42180568185, 5720993700540, 1088246094845838, 291276119631119408, 109983236494820652007, 58741463418913578672779, 44466318283501559718838424, 47771843216826858235974983400, 72930986725295232949801895385998
Offset: 0

Views

Author

Paul D. Hanna, Nov 16 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 + 8*x^2 + 128*x^3 + 3259*x^4 + 120082*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^4*x^2*A(2*x)^2/2!*exp(-2^2*x*A(2*x)) + 3^6*x^3*A(3*x)^3/3!*exp(-3^2*x*A(3*x)) + 4^8*x^4*A(4*x)^4/4!*exp(-4^2*x*A(4*x)) + 5^10*x^5*A(5*x)^5/5!*exp(-5^2*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A219265, A218681, A218672, A219184, A217900.

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=0,25,print1(a(n),", "))

A219265 O.g.f. satisfies: A(x) = Sum_{n>=0} A(n^2*x)^n * (n^2*x)^n/n! * exp(-n^2*x*A(n^2*x)).

Original entry on oeis.org

1, 1, 8, 160, 6918, 609469, 106947753, 37651271215, 26931993643529, 39243099256414069, 116654228928308598913, 710224935200206160129234, 8867331728829780268501045551, 227187317486051730833557991305666, 11969414396907448200529521385052444890
Offset: 0

Views

Author

Paul D. Hanna, Nov 16 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 + 8*x^2 + 160*x^3 + 6918*x^4 + 609469*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^4*x^2*A(2^2*x)^2/2!*exp(-2^2*x*A(2^2*x)) + 3^6*x^3*A(3^2*x)^3/3!*exp(-3^2*x*A(3^2*x)) + 4^8*x^4*A(4^2*x)^4/4!*exp(-4^2*x*A(4^2*x)) + 5^10*x^5*A(5^2*x)^5/5!*exp(-5^2*x*A(5^2*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A219264, A218681, A218672, A219184, A217900.

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^2*x)^k/k!*exp(-k^2*x*subst(A,x,k^2*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A219343 O.g.f. satisfies: A(x) = Sum_{n>=0} A(n*x)^n * (n^3*x)^n/n! * exp(-n^3*x*A(n*x)).

Original entry on oeis.org

1, 1, 32, 3183, 650929, 226009218, 119298668857, 89086101638412, 89480710389500666, 116491795770107486363, 191172400354899371561288, 387419202671209086703674709, 956322827450633453264262285623, 2859815748552720894795327258080881, 10430012061189048036456303441601971435
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 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 + 32*x^2 + 3183*x^3 + 650929*x^4 + 226009218*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^6*x^2*A(2*x)^2/2!*exp(-2^3*x*A(2*x)) + 3^9*x^3*A(3*x)^3/3!*exp(-3^3*x*A(3*x)) + 4^12*x^4*A(4*x)^4/4!*exp(-4^3*x*A(4*x)) + 5^15*x^5*A(5*x)^5/5!*exp(-5^3*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A219264, A219265, A218681, A218672, A219184, A219342, A217900.

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=0,25,print1(a(n),", "))
Showing 1-8 of 8 results.

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