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.

Previous Showing 11-20 of 25 results. Next

A218676 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(5*n)/n! * exp(-n*x*A(n*x)^5).

Original entry on oeis.org

1, 1, 6, 71, 1311, 34146, 1207717, 57298282, 3653975784, 316252925221, 37596625187796, 6206102367103899, 1434418185304457039, 466995106832397752352, 215051811411620578152401, 140491107719613466192347681, 130481943378389095603359529403
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 + 6*x^2 + 71*x^3 + 1311*x^4 + 34146*x^5 + 1207717*x^6 +...
where
A(x) = 1 + x*A(x)^5*exp(-x*A(x)^5) + 2^2*x^2*A(2*x)^10/2!*exp(-2*x*A(2*x)^5) + 3^3*x^3*A(3*x)^15/3!*exp(-3*x*A(3*x)^5) + 4^4*x^4*A(4*x)^20/4!*exp(-4*x*A(4*x)^5) + 5^5*x^5*A(5*x)^25/5!*exp(-5*x*A(5*x)^5) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A218673, A218674, A218675.
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^5,x,k*x)^k/k!*exp(-k*x*subst(A^5,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.

A209276 O.g.f. satisfies: A(x) = Sum_{n>=0} (n+2)^n * x^n * A((n+2)*x)^n/n! * exp(-(n+2)*x*A((n+2)*x)).

Original entry on oeis.org

1, 1, 6, 133, 9403, 2065969, 1400088539, 2908156231705, 18410003437367130, 353588715425938097698, 20534146782689861283550052, 3596867485365965032072729708845, 1897112888731795684931545113460297299, 3009299517165127420220975531888408947667944
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 + 6*x^2 + 133*x^3 + 9403*x^4 + 2065969*x^5 +...
where
A(x) = exp(-2*x*A(2*x)) + 3*x*A(3*x)*exp(-3*x*A(3*x)) + 4^2*x^2*A(4*x)^2/2!*exp(-4*x*A(4*x)) + 5^3*x^3*A(5*x)^3/3!*exp(-5*x*A(5*x)) + 6^4*x^4*A(6*x)^4/4!*exp(-6*x*A(6*x)) + 7^5*x^5*A(7*x)^5/5!*exp(-7*x*A(7*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A193363, A209277.

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), ", "))

A209277 O.g.f. satisfies: A(x) = Sum_{n>=0} (n+3)^n * x^n * A((n+3)*x)^n/n! * exp(-(n+3)*x*A((n+3)*x)).

Original entry on oeis.org

1, 1, 8, 321, 57879, 45643415, 154158595175, 2190765237132015, 129241431881731600186, 31339180791153421540163500, 31011964321205837200260130287298, 124581202469689320858858825068619255535, 2023924731754579903607034623889070335771466703
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 + 8*x^2 + 321*x^3 + 57879*x^4 + 45643415*x^5 +...
where
A(x) = exp(-3*x*A(3*x)) + 4*x*A(4*x)*exp(-4*x*A(4*x)) + 5^2*x^2*A(5*x)^2/2!*exp(-5*x*A(5*x)) + 6^3*x^3*A(6*x)^3/3!*exp(-6*x*A(6*x)) + 7^4*x^4*A(7*x)^4/4!*exp(-7*x*A(7*x)) + 8^5*x^5*A(8*x)^5/5!*exp(-8*x*A(8*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A193363, A209276.

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

A216246 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, 1, 2, 7, 20, 62, 228, 867, 3474, 14864, 67101, 316028, 1540782, 7792810, 40669011, 218087867, 1201778736, 6792071574, 39309770969, 232718797430, 1407837227275, 8694994837673, 54771751869237, 351643945379956, 2299330292987022, 15302662859459784
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2013

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 + x^2 + 2*x^3 + 7*x^4 + 20*x^5 + 62*x^6 + 228*x^7 +...
where
A(x) = 1 + x*A(x^2)*exp(-x*A(x^2)) + 2^2*x^2*A(2*x^2)^2/2!*exp(-2*x*A(2*x^2)) + 3^3*x^3*A(3*x^2)^3/3!*exp(-3*x*A(3*x^2)) + 4^4*x^4*A(4*x^2)^4/4!*exp(-4*x*A(4*x^2)) + 5^5*x^5*A(5*x^2)^5/5!*exp(-5*x*A(5*x^2)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A216247, A218672, 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*x^2)^k/k!*exp(-k*x*subst(A, x, k*x^2)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
Previous Showing 11-20 of 25 results. Next

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