A382039 -id:A382039 - cp's OEIS frontend

A382040 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - xexp(4*x)) ).

1, 1, 12, 198, 4912, 163120, 6796224, 341366704, 20088997632, 1356164492544, 103333898644480, 8773563043734016, 821474949840482304, 84093840447771701248, 9344359942839980900352, 1120159940123276849141760, 144096985208727744665288704, 19800296439825918648654561280

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A213644, A379690, A382039.

Cf. A366234, A382031.

a(n) = sum(k=0, n, (4*k)^(n-k)*(n+k)!/(k!*(n-k)!))/(n+1);

E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2*exp(4*x*A(x)).

a(n) = (1/(n+1)) * Sum_{k=0..n} (4*k)^(n-k) * (n+k)!/(k! * (n-k)!).

A382042 E.g.f. A(x) satisfies A(x) = exp(xC(xA(x)^3)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 3, 37, 733, 20181, 714541, 30903769, 1579206441, 93099946249, 6219777779641, 464382363698661, 38319628830696973, 3463058939163189133, 340172205752538636933, 36087128101110502864561, 4111807211977470782285521, 500807663307856030823859729, 64931674940413564774656214513

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A161629, A382029.

Cf. A000108, A212917, A382039.

a(n) = if(n==0, 1, n!*sum(k=0, n-1, (3*k+1)^(n-k-1)*binomial(n+k, k)/((n+k)*(n-k-1)!)));

Let F(x) be the e.g.f. of A382039. F(x) = log(A(x))/x = C(x*A(x)^3).
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(3*x)) ) ).
a(n) = n! * Sum_{k=0..n-1} (3*k+1)^(n-k-1) * binomial(n+k,k)/((n+k) * (n-k-1)!) for n > 0.

cp's OEIS Frontend

A382040 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - xexp(4*x)) ).

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A382042 E.g.f. A(x) satisfies A(x) = exp(xC(xA(x)^3)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

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cp's OEIS Frontend

A382040 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(4*x)) ).

Views

Author

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Crossrefs

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PARI

Formula

A382042 E.g.f. A(x) satisfies A(x) = exp(x*C(x*A(x)^3)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Views

Author

Keywords

Crossrefs

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PARI

Formula

A382040 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - xexp(4*x)) ).

A382042 E.g.f. A(x) satisfies A(x) = exp(xC(xA(x)^3)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.