A162695 -id:A162695 - cp's OEIS frontend

A360474 E.g.f. satisfies A(x) = exp( x * A(x)^2 * exp(x * A(x)^2) ).

1, 1, 7, 94, 1921, 53036, 1849789, 78070462, 3869909537, 220427550712, 14188370562901, 1018570771664546, 80692202644742737, 6992855583524143204, 658076908751441373965, 66833181471569822199886, 7285736943975575120653249, 848589321771735983890457072

Offset: 0

Seiichi Manyama, Feb 08 2023

nonn

Cf. A162695, A360473.

Join[{1}, Table[Sum[k^(n-k) * (2*n+1)^(k-1) * Binomial[n,k], {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Feb 17 2023 *)

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

a(n) = Sum_{k=0..n} k^(n-k) * (2*n+1)^(k-1) * binomial(n,k).

a(n) ~ 2^(n - 1/2) * n^(n-1) * s^(2*n + 1) * log(s)^(n + 1/2) / (sqrt(1 + 2*log(s) - 4*log(s)^2) * exp(n) * (1 - 2*log(s))^n), where s = 1.473428520956658037187728756446912746332041803082... is the root of the equation 2*log(s)*(1 + LambertW(log(s))) = 1. - Vaclav Kotesovec, Feb 17 2023

A380425 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) * exp(x * A(x)^(1/2)) ).

Original entry on oeis.org

1, 2, 12, 116, 1592, 28472, 630028, 16649348, 512197456, 17993496176, 711065689364, 31231930472492, 1509776777566648, 79670350504209896, 4557716010219325468, 280992142281969312548, 18574365176584473753248, 1310583528463442480750048, 98318677221689347734929956

Offset: 0

Seiichi Manyama, Jan 24 2025

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..360

Cf. A162695, A380426, A380427.

Cf. A360474, A380406.

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

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A162695.

a(n) = 2 * Sum_{k=0..n} k^(n-k) * (n+2)^(k-1) * binomial(n,k).

A380426 E.g.f. A(x) satisfies A(x) = exp( 3 * x * A(x)^(1/3) * exp(x * A(x)^(1/3)) ).

Original entry on oeis.org

1, 3, 21, 225, 3309, 62223, 1430235, 38940681, 1227116409, 43970226651, 1766653847079, 78696970239165, 3850658628709941, 205350233796536871, 11856632842453069491, 736988120901130761297, 49073265311942508067185, 3485242354486865203370931, 263004127262410708414755135

Offset: 0

Seiichi Manyama, Jan 24 2025

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..360

Cf. A162695, A380425, A380427.

Cf. A380407.

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

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A162695.

a(n) = 3 * Sum_{k=0..n} k^(n-k) * (n+3)^(k-1) * binomial(n,k).

A380427 E.g.f. A(x) satisfies A(x) = exp( -x/A(x) * exp(x/A(x)) ).

Original entry on oeis.org

1, -1, -3, -19, -211, -3301, -66581, -1643587, -47986247, -1617313033, -61796668969, -2639583958111, -124635062782187, -6446216079166189, -362427406400015165, -22008570202561166491, -1435560535563493528591, -100100185675457848764433, -7430481272601559979203409

Offset: 0

Seiichi Manyama, Jan 24 2025

sign

Seiichi Manyama, Table of n, a(n) for n = 0..360

Cf. A162695, A380425, A380426.

Cf. A357247.

E.g.f.: 1/B(x), where B(x) is the e.g.f. of A162695.

a(n) = -Sum_{k=1..n} k^(n-k) * (n-1)^(k-1) * binomial(n,k) for n > 0.

A380880 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3xexp(x)) ).

Original entry on oeis.org

1, 3, 33, 657, 19317, 756663, 37153071, 2196991317, 152107121481, 12074764795947, 1081507189545219, 107911010079715857, 11871250914793342797, 1427601609871824349407, 186326851375925627135127, 26232637698244127999077677, 3962908338833364902518738449, 639433805204122165558890771027

Offset: 0

Seiichi Manyama, Feb 07 2025

nonn

Index entries for reversions of series

Cf. A162695, A380879.

Cf. A380881.

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

E.g.f. A(x) satisfies A(x) = exp(3 * x * A(x) * exp(x * A(x))).
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A380881.
a(n) = 3 * Sum_{k=0..n} k^(n-k) * (3*n+3)^(k-1) * binomial(n,k).

A380879 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2xexp(x)) ).

Original entry on oeis.org

1, 2, 16, 230, 4888, 138442, 4916140, 210270734, 10530743632, 604747157138, 39185881490644, 2828691317839510, 225137088955561144, 19588316964130880474, 1849745928662841982588, 188421660506420000503838, 20594905554562935801454240, 2404374864844251715105658146

Offset: 0

Seiichi Manyama, Feb 07 2025

nonn

Index entries for reversions of series

Cf. A162695, A380880.

Cf. A360474.

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

E.g.f. A(x) satisfies A(x) = exp(2 * x * A(x) * exp(x * A(x))).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A360474.
a(n) = 2 * Sum_{k=0..n} k^(n-k) * (2*n+2)^(k-1) * binomial(n,k).

A380881 E.g.f. A(x) satisfies A(x) = exp( x * A(x)^3 * exp(x * A(x)^3) ).

Original entry on oeis.org

1, 1, 9, 163, 4541, 171781, 8231395, 478055299, 32642065433, 2562896897353, 227510655792191, 22533214047347455, 2463465770439307045, 294676777871863052173, 38284087227668033391515, 5368383942726216941810971, 808133883137288259018215345, 129988823008132636178027546257

Offset: 0

Seiichi Manyama, Feb 07 2025

nonn

Index entries for reversions of series

Cf. A162695, A360474.

Cf. A362656, A380880.

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

E.g.f.: ( (1/x) * Series_Reversion( x * exp(-3*x*exp(x)) ) )^(1/3).

a(n) = Sum_{k=0..n} k^(n-k) * (3*n+1)^(k-1) * binomial(n,k).

A380972 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-xexp(2x)) ).

Original entry on oeis.org

1, 1, 7, 76, 1237, 26816, 728899, 23866816, 915129961, 40237778944, 1996402790431, 110351882157056, 6725593733125117, 448106469169905664, 32404532970216803803, 2527793703574203252736, 211589448225820679029969, 18917558526854862344290304, 1799285901282568752019291063

Offset: 0

Seiichi Manyama, Feb 10 2025

nonn

Index entries for reversions of series

Cf. A162695, A380973.

Cf. A380879.

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

E.g.f. A(x) satisfies A(x) = exp( x*A(x) * exp(2*x*A(x)) ).

a(n) = Sum_{k=0..n} (2*k)^(n-k) * (n+1)^(k-1) * binomial(n,k).

A380973 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-xexp(3x)) ).

Original entry on oeis.org

1, 1, 9, 115, 2213, 56781, 1825735, 70718383, 3207565737, 166830072409, 9791107408331, 640182529765395, 46152280917472669, 3637314366894167077, 311129703773921407887, 28708644100373375591191, 2842495895373573092038865, 300611288206029730901431473, 33820062046972635799385887123

Offset: 0

Seiichi Manyama, Feb 10 2025

nonn

Index entries for reversions of series

Cf. A162695, A380972.

Cf. A380880.

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

E.g.f. A(x) satisfies A(x) = exp( x*A(x) * exp(3*x*A(x)) ).

a(n) = Sum_{k=0..n} (3*k)^(n-k) * (n+1)^(k-1) * binomial(n,k).

cp's OEIS Frontend

A360474 E.g.f. satisfies A(x) = exp( x * A(x)^2 * exp(x * A(x)^2) ).

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A380425 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) * exp(x * A(x)^(1/2)) ).

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A380426 E.g.f. A(x) satisfies A(x) = exp( 3 * x * A(x)^(1/3) * exp(x * A(x)^(1/3)) ).

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A380427 E.g.f. A(x) satisfies A(x) = exp( -x/A(x) * exp(x/A(x)) ).

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A380880 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x*exp(x)) ).

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PARI

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A380879 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x*exp(x)) ).

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A380881 E.g.f. A(x) satisfies A(x) = exp( x * A(x)^3 * exp(x * A(x)^3) ).

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A380972 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x*exp(2*x)) ).

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A380973 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x*exp(3*x)) ).

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Programs

PARI

Formula

A380880 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3xexp(x)) ).

A380879 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2xexp(x)) ).

A380972 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-xexp(2x)) ).

A380973 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-xexp(3x)) ).