A382000 -id:A382000 - 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

A382001 E.g.f. A(x) satisfies A(x) = 1 + xexp(2x)*A(x)^6.

Original entry on oeis.org

1, 1, 16, 462, 20672, 1261400, 97728672, 9190016416, 1016963389696, 129485497897728, 18648682990461440, 2997567408967391744, 531985786683988512768, 103321584851593487961088, 21798243872991807130685440, 4964302861788729054456729600, 1213816740632458735310221672448

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

In general, if k>1 and e.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^k, then a(n) ~ sqrt(k) * sqrt(1 + LambertW(2*(k-1)^(k-1)/k^k)) * 2^n * n^(n-1) / ((k-1)^(3/2) * exp(n) * LambertW(2*(k-1)^(k-1)/k^k)^n). - Vaclav Kotesovec, Mar 22 2025

Cf. A336950, A381997, A381998, A381999, A382000.

Cf. A002295, A381989.

a(n) = n!*sum(k=0, n, (2*k)^(n-k)*binomial(6*k+1, k)/((6*k+1)*(n-k)!));

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A002295(k)/(n-k)!.

a(n) ~ sqrt(3*(1 + LambertW(3125/23328))) * 2^(n + 1/2) * n^(n-1) / (5^(3/2) * exp(n) * LambertW(3125/23328)^n). - Vaclav Kotesovec, Mar 22 2025

A381998 E.g.f. A(x) satisfies A(x) = 1 + xexp(2x)*A(x)^2.

Original entry on oeis.org

1, 1, 8, 90, 1472, 31920, 865152, 28197904, 1075122176, 46976064768, 2315080816640, 127068467480064, 7688296957870080, 508450036968779776, 36490818871396499456, 2824787199565881477120, 234622076533699738861568, 20813348299168251651883008, 1964063064959266899440959488

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A336950, A381997, A381999, A382000, A382001.

Cf. A000108, A295238, A379885.

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

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A000108(k)/(n-k)!.
From Vaclav Kotesovec, Mar 22 2025: (Start)
E.g.f.: 2/(1 + sqrt(1 - 4*exp(2*x)*x)).
a(n) ~ sqrt(1 + LambertW(1/2)) * 2^(n + 1/2) * n^(n-1) / (exp(n) * LambertW(1/2)^n). (End)

A381999 E.g.f. A(x) satisfies A(x) = 1 + xexp(2x)*A(x)^3.

Original entry on oeis.org

1, 1, 10, 156, 3656, 115400, 4595232, 221281312, 12510826624, 812633118336, 59642105050880, 4881685773730304, 440905471531302912, 43559980305765793792, 4673231270870843441152, 541042726968231082967040, 67236501012517546330062848, 8927220151967826907452440576

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A336950, A381997, A381998, A382000, A382001.

Cf. A001764, A364983, A371318.

a(n) = n!*sum(k=0, n, (2*k)^(n-k)*binomial(3*k+1, k)/((3*k+1)*(n-k)!));

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A001764(k)/(n-k)!.

a(n) ~ sqrt(3*(1 + LambertW(8/27))) * 2^(n - 3/2) * n^(n-1) / (exp(n) * LambertW(8/27)^n). - Vaclav Kotesovec, Mar 22 2025

A381986 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)^2), where B(x) = 1 + xB(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 2, 17, 388, 14329, 727206, 46984729, 3689119624, 341097752657, 36302764864330, 4371463743828481, 587606216836328460, 87219196719691250185, 14168990447072685567214, 2500554381188629649979593, 476391652257266128440376336, 97447147561230881896398507553

Offset: 0

Seiichi Manyama, Mar 11 2025

nonn

Cf. A381984, A381985.

Cf. A001764, A002294, A382000.

a(n) = n!*sum(k=0, n, (2*k+1)^(n-k)*binomial(5*k+1, k)/((5*k+1)*(n-k)!));

Let F(x) be the e.g.f. of A382000. F(x) = B(x*A(x)^2) = exp( 1/3 * Sum_{k>=1} binomial(3*k,k) * (x*A(x)^2)^k/k ).

a(n) = n! * Sum_{k=0..n} (2*k+1)^(n-k) * A002294(k)/(n-k)!.

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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A382001 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^6.

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A381998 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^2.

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A381999 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^3.

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A381986 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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

A382001 E.g.f. A(x) satisfies A(x) = 1 + xexp(2x)*A(x)^6.

A381998 E.g.f. A(x) satisfies A(x) = 1 + xexp(2x)*A(x)^2.

A381999 E.g.f. A(x) satisfies A(x) = 1 + xexp(2x)*A(x)^3.

A381986 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)^2), where B(x) = 1 + xB(x)^3 is the g.f. of A001764.