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 1-7 of 7 results.

A221096 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(2*n))^n/n!.

Original entry on oeis.org

1, 1, 4, 42, 768, 19460, 637200, 25724916, 1233957312, 68591031120, 4338982958400, 307907317681920, 24229505587541760, 2094548798610726432, 197370092438311892736, 20140182770328963216000, 2213078753956025271214080, 260601290312643875434817280
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 19460*x^5/5! +...
where A(x) satisfies:
A(x) = 1 + log(1 + x*A(x)^2) + log(1 + x*A(x)^4)^2/2! + log(1 + x*A(x)^6)^3/3! +...
The e.g.f. also satisfies:
A(x) = 1 + A(x)^2*x + A(x)^4*(A(x)^4-1)*x^2/2! + A(x)^6*(A(x)^6-1)*(A(x)^6-2)*x^3/3! + A(x)^8*(A(x)^8-1)*(A(x)^8-2)*(A(x)^8-3)*x^4/4! +...+ binomial(A(x)^(2*n), n)*x^n +...

Crossrefs

Cf. A189981, A221097, A221098, A221099.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+x*(A+x*O(x^n))^(2*m))^m/m!)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, binomial((A+x*O(x^n))^(2*m), m)*x^m)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*(A+x*O(x^n))^(2*m*k))*x^m/m!)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. also satisfies:
(1) A(x) = Sum_{n>=0} binomial(A(x)^(2*n), n) * x^n.
(2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(2*n*k)/n!.

A221097 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(3*n))^n/n!.

Original entry on oeis.org

1, 1, 6, 90, 2328, 84660, 3972060, 229176654, 15712089120, 1248343353216, 112832687750400, 11437476445244520, 1285433373363701760, 158682294244352658312, 21349655111889802728576, 3110218068324341815470000, 487862693943123978219847680, 81999755541558838752430348800
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 90*x^3/3! + 2328*x^4/4! + 84660*x^5/5! +...
where A(x) satisfies:
A(x) = 1 + log(1 + x*A(x)^3) + log(1 + x*A(x)^6)^2/2! + log(1 + x*A(x)^9)^3/3! +...
The e.g.f. also satisfies:
A(x) = 1 + A(x)^3*x + A(x)^6*(A(x)^6-1)*x^2/2! + A(x)^9*(A(x)^9-1)*(A(x)^9-2)*x^3/3! + A(x)^12*(A(x)^12-1)*(A(x)^12-2)*(A(x)^12-3)*x^4/4! +...+ binomial(A(x)^(3*n), n)*x^n +...

Crossrefs

Cf. A189981, A221096, A221098, A221099.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+x*(A+x*O(x^n))^(3*m))^m/m!)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, binomial((A+x*O(x^n))^(3*m), m)*x^m)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*(A+x*O(x^n))^(3*m*k))*x^m/m!)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. also satisfies:
(1) A(x) = Sum_{n>=0} binomial(A(x)^(3*n), n) * x^n.
(2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(3*n*k)/n!.

A221098 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(4*n))^n/n!.

Original entry on oeis.org

1, 1, 8, 156, 5184, 243280, 14742240, 1097403552, 97012667136, 9936480419424, 1157549828855040, 151193318253405120, 21890302973632558080, 3480525852596442818688, 603034041051994953483264, 113109668528001746742489600, 22839699845167989485088522240
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 156*x^3/3! + 5184*x^4/4! + 243280*x^5/5! +...
where A(x) satisfies:
A(x) = 1 + log(1 + x*A(x)^4) + log(1 + x*A(x)^8)^2/2! + log(1 + x*A(x)^12)^3/3! +...
The e.g.f. also satisfies:
A(x) = 1 + A(x)^4*x + A(x)^8*(A(x)^8-1)*x^2/2! + A(x)^12*(A(x)^12-1)*(A(x)^12-2)*x^3/3! + A(x)^16*(A(x)^16-1)*(A(x)^16-2)*(A(x)^16-3)*x^4/4! +...+ binomial(A(x)^(4*n), n)*x^n +...

Crossrefs

Cf. A189981, A221096, A221097, A221099.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+x*(A+x*O(x^n))^(4*m))^m/m!)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, binomial((A+x*O(x^n))^(4*m), m)*x^m)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*(A+x*O(x^n))^(4*m*k))*x^m/m!)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. also satisfies:
(1) A(x) = Sum_{n>=0} binomial(A(x)^(4*n), n) * x^n.
(2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(4*n*k)/n!.

A221099 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^(5*n))^n/n!.

Original entry on oeis.org

1, 1, 10, 240, 9720, 556400, 41153220, 3737360130, 402876727680, 50302825722720, 7141958361129600, 1136668023900846360, 200486825731741824000, 38826473000115470677800, 8192096172894406564646400, 1870885111733841408594984000, 459893703431651653070494156800
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 10*x^2/2! + 240*x^3/3! + 9720*x^4/4! + 556400*x^5/5! +...
where A(x) satisfies:
A(x) = 1 + log(1 + x*A(x)^5) + log(1 + x*A(x)^10)^2/2! + log(1 + x*A(x)^15)^3/3! +...
The e.g.f. also satisfies:
A(x) = 1 + A(x)^5*x + A(x)^10*(A(x)^10-1)*x^2/2! + A(x)^15*(A(x)^15-1)*(A(x)^15-2)*x^3/3! + A(x)^20*(A(x)^20-1)*(A(x)^20-2)*(A(x)^20-3)*x^4/4! +...+ binomial(A(x)^(5*n), n)*x^n +...

Crossrefs

Cf. A189981, A221096, A221097, A221098.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+x*(A+x*O(x^n))^(5*m))^m/m!)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, binomial((A+x*O(x^n))^(5*m), m)*x^m)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*(A+x*O(x^n))^(5*m*k))*x^m/m!)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. also satisfies:
(1) A(x) = Sum_{n>=0} binomial(A(x)^(5*n), n) * x^n.
(2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(5*n*k)/n!.

A221101 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(n*x))^n/n!.

Original entry on oeis.org

1, 1, 2, 12, 144, 3160, 118380, 7174188, 692356896, 104696597808, 24680489921280, 9010186432576560, 5073501307520289600, 4385657278007399474496, 5797249519065509217375936, 11674185903250032386477342880, 35692663320428574506107140979200
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 144*x^4/4! + 3160*x^5/5! +...
where
A(x) = 1 + log(1 + x*A(x)) + log(1 + x*A(2*x))^2/2! + log(1 + x*A(3*x))^3/3! + log(1 + x*A(4*x))^4/4! +...

Crossrefs

Cf. A189981, A221100.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+x*subst(A,x,m*x+x*O(x^n)))^m/m!)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*subst(A,x,k*x+x*O(x^n))^m)*x^m/m!)); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(k*x)^n/n!.

A224797 E.g.f. satisfies: A(x) = Sum_{n>=0} (exp(x*A(x)^n) - 1)^n / n!.

Original entry on oeis.org

1, 1, 4, 35, 503, 10207, 268865, 8731102, 337630732, 15165277773, 776576049655, 44683002944571, 2855602714004089, 200794017101260026, 15413426272667102594, 1283152929854467388195, 115198576226248396583523, 11099504126776462035978911
Offset: 0

Views

Author

Paul D. Hanna, Apr 18 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 35*x^3/3! + 503*x^4/4! + 10207*x^5/5! +...
where
A(x) = 1 + (exp(x*A(x)) - 1) + (exp(x*A(x)^2) - 1)^2/2! + (exp(x*A(x)^3) - 1)^3/3! + (exp(x*A(x)^4) - 1)^4/4! + (exp(x*A(x)^5) - 1)^5/5! +...

Crossrefs

Cf. A189981.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (exp(x*A^m +x*O(x^n))-1)^m/m!)); n!*polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))
  • PARI
    {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling2(m, k)*(A+x*O(x^n))^(m*k))*x^m/m!)); n!*polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

Formula

E.g.f. satisfies: A(x) = Sum_{n>=0} Sum_{k=0..n} Stirling2(n, k)*A(x)^(n*k) * x^n/n!.

A214930 E.g.f. satisfies: A(x) = Sum_{n>=0} 1/n! * Product_{k=1..n} log(1 + x*A(x)^k).

Original entry on oeis.org

1, 1, 2, 9, 66, 650, 8250, 127519, 2318876, 48626556, 1154334060, 30589513350, 895415799960, 28693464851688, 999009599484624, 37554576369815400, 1516080931559327280, 65418533528228549744, 3004726893339734134128, 146370356574519380115240
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 66*x^4/4! + 650*x^5/5! +...
where
A(x) = 1 + log(1+x*A(x)) + log(1+x*A(x))*log(1+x*A(x)^2)/2! + log(1+x*A(x))*log(1+x*A(x)^2)*log(1+x*A(x)^3)/3! + log(1+x*A(x))*log(1+x*A(x)^2)*log(1+x*A(x)^3)*log(1+x*A(x)^4)/4! +...

Crossrefs

Cf. A189981, A033917.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,prod(k=1,m,log(1+x*A^k+x*O(x^n)))/m!));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
Showing 1-7 of 7 results.

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