E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 5^n * (exp(x)-1)^n * x^n / n!.

E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 5^n * (exp(x)-1)^n * x^(n-1) / n! ).

{a(n)=local(A=x); A = x + sum(m=1, n, Dx(m-1, 5^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}

{a(n)=local(A=x+x^2+x*O(x^n)); A = x*exp(sum(m=1, n, Dx(m-1, 5^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}

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E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 5^n*(exp(x)-1)^n*x^n / n!.

E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 5^n*(exp(x)-1)^n * x^(n-1) / n! ).

(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, 5^m*(exp(x+x*O(x^n))-1)^m*x^m/m!)); n!*polcoeff(A, n)}

for(n=1, 25, print1(a(n), ", "))

(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, 5^m*(exp(x+x*O(x^n))-1)^m*x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}

for(n=1, 25, print1(a(n), ", "))

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E.g.f.: Series_Reversion( 6*x - 5*x*exp(x) ).

1, 10, 315, 16520, 1212775, 114465780, 13204213435, 1800094703440, 283154358503295, 50478562633826300, 10057594831485171355, 2214859039031666012760, 534202513174577053611415, 140048168049127802257998820, 39652657811418543065286846075, 12058716801545122639605896216480, 3920065606375381780966255214893135

O.g.f.: x * Sum_{n>=0} 5^n / (6 - n*x)^(n+1).

E.g.f.: A(x) = x + 10*x^2/2! + 315*x^3/3! + 16520*x^4/4! + 1212775*x^5/5! +...

(PARI) {a(n) = local(A=x); A = serreverse(6*x - 5*x*exp(x +x*O(x^n) )); n!*polcoeff(A, n)}

Cf. A053492, A259063, A259064, A259065.

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allocated for Paul D. Hanna

E.g.f.: Series_Reversion( 6*x - 5*x*exp(x) ).

1, 10, 315, 16520, 1212775, 114465780, 13204213435, 1800094703440, 283154358503295, 50478562633826300, 10057594831485171355, 2214859039031666012760, 534202513174577053611415, 140048168049127802257998820, 39652657811418543065286846075, 12058716801545122639605896216480, 3920065606375381780966255214893135

1,2

O.g.f.: x * Sum_{n>=0} 5^n / (6 - n*x)^(n+1).

E.g.f.: A(x) = x + 10*x^2/2! + 315*x^3/3! + 16520*x^4/4! + 1212775*x^5/5! +...

where A(6*x - 5*x*exp(x)) = x.

Also we have the related infinite series.

O.g.f.: F(x) = x + 10*x^2 + 315*x^3 + 16520*x^4 + 1212775*x^5 +...

where

F(x)/x = 1/6 + 5/(6-x)^2 + 5^2/(6-2*x)^3 + 5^3/(6-3*x)^4 + 5^4/(6-4*x)^5 +...

(PARI) {a(n) = local(A=x); A = serreverse(6*x - 5*x*exp(x +x*O(x^n) )); n!*polcoeff(A, n)}

for(n=1, 20, print1(a(n), ", "))

Cf. A053492, A259063, A259064, A259065.

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nonn

Paul D. Hanna, Jun 17 2015

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allocated for Paul D. Hanna

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