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
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 +...
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{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),", "))
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{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),", "))
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{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),", "))
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
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 +...
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{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),", "))
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{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),", "))
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{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),", "))
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
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 +...
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{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),", "))
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{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),", "))
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{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),", "))
Showing 1-3 of 3 results.