A381983 -id:A381983 - cp's OEIS frontend

A381997 E.g.f. A(x) satisfies A(x) = 1 + xexp(2x)*A(x)^4.

1, 1, 12, 240, 7328, 303400, 15904032, 1010252320, 75442821120, 6478112692224, 628915387166720, 68121797696449024, 8144844724723482624, 1065508614975814537216, 151392999512027274215424, 23217165210450099377479680, 3822334349865128121165283328, 672407573328393115218009063424

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A336950, A381998, A381999, A382000, A382001.

Cf. A002293, A364987, A381983.

A381997 := proc(n)
        n!*add((2*k)^(n-k)*binomial(4*k+1,k)/(4*k+1)/(n-k)!,k=0..n) ;
end proc:
seq(A381997(n),n=0..60) ;  # R. J. Mathar, Mar 12 2025

a(n) = n!*sum(k=0, n, (2*k)^(n-k)*binomial(4*k+1, k)/((4*k+1)*(n-k)!));

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A002293(k)/(n-k)!.

a(n) ~ 2^(n+1) * n^(n-1) * sqrt(1 + LambertW(27/128)) / (3^(3/2) * exp(n) * LambertW(27/128)^n). - Vaclav Kotesovec, Mar 22 2025

A381982 E.g.f. A(x) satisfies A(x) = exp(x) * C(xA(x)), where C(x) = 1 + xC(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 2, 11, 139, 2829, 78981, 2802163, 120667667, 6113752025, 356342305465, 23488872131871, 1727770084512495, 140302645206245701, 12466960491079733237, 1203253101643330233707, 125351056198801059896491, 14019427299278115378992049, 1675439381194882102492648305

Offset: 0

Seiichi Manyama, Mar 11 2025

nonn

Cf. A349640, A381983.

Cf. A000108, A001764, A161629, A364983.

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

Let F(x) be the e.g.f. of A364983. F(x) = C(x*A(x)) = exp( 1/2 * Sum_{k>=1} binomial(2*k,k) * (x*A(x))^k/k ).

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * A001764(k)/(n-k)!.

cp's OEIS Frontend

A381997 E.g.f. A(x) satisfies A(x) = 1 + xexp(2x)*A(x)^4.

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A381982 E.g.f. A(x) satisfies A(x) = exp(x) * C(xA(x)), where C(x) = 1 + xC(x)^2 is the g.f. of A000108.

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cp's OEIS Frontend

A381997 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^4.

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Author

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Crossrefs

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Maple

PARI

Formula

A381982 E.g.f. A(x) satisfies A(x) = exp(x) * C(x*A(x)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

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Formula

A381997 E.g.f. A(x) satisfies A(x) = 1 + xexp(2x)*A(x)^4.

A381982 E.g.f. A(x) satisfies A(x) = exp(x) * C(xA(x)), where C(x) = 1 + xC(x)^2 is the g.f. of A000108.