A088690 -id:A088690 - cp's OEIS frontend

A379936 E.g.f. A(x) satisfies A(x) = 1/( exp(-x*A(x)^(1/2)) - x )^2.

1, 4, 30, 344, 5400, 108492, 2667952, 77811120, 2629399680, 101122817300, 4363964377344, 208925612290056, 10992411683169280, 630611992509716700, 39182624685283891200, 2621745777377998537568, 187969244952968687812608, 14377545994804829244970020

Offset: 0

Seiichi Manyama, Jan 06 2025

nonn

Cf. A379933, A379934.

Cf. A088690.

my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x*exp(-x)/(1+x))/x)^2))

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

E.g.f.: ( (1/x) * Series_Reversion( x*exp(-x)/(1+x) ) )^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A088690.
a(n) = 2 * n! * Sum_{k=0..n} (n+2)^(k-1) * binomial(n+2,n-k)/k!.

A380646 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x)/(1 + x)^2 ).

Original entry on oeis.org

1, 4, 46, 932, 27568, 1080432, 52916176, 3115326496, 214470890496, 16914853191680, 1504252282653184, 148956086481767424, 16256865070022066176, 1938988214539948730368, 250943399365390735104000, 35026523834624205803491328, 5245178283068781060488298496, 838841884254236846183525646336

Offset: 0

Seiichi Manyama, Feb 06 2025

nonn

Index entries for reversions of series

Cf. A088690, A380647, A380648.
Cf. A065866, A377829.
Cf. A097629, A380828.
Cf. A377892.

nmax=18; CoefficientList[(1/x)InverseSeries[Series[x*Exp[-2*x]/(1 + x)^2 ,{x,0,nmax}]],x]Range[0,nmax-1]! (* Stefano Spezia, Feb 06 2025 *)

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

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^2 * exp(2 * x * A(x)).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A377892.
a(n) = 2 * n! * Sum_{k=0..n} (2*n+2)^(k-1) * binomial(2*n+2,n-k)/k!.

A380647 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x)/(1 + x)^3 ).

Original entry on oeis.org

1, 6, 105, 3246, 146637, 8780688, 657224901, 59140486800, 6223651526457, 750357182131200, 102014741343847329, 15443915464974191616, 2576937457466957107845, 469914373917914931984384, 92982800086882512621716925, 19843243096453465663599962112, 4543276116844426827394718716401

Offset: 0

Seiichi Manyama, Feb 06 2025

nonn

Index entries for reversions of series

Cf. A088690, A380646, A380648.
Cf. A370055, A377830.
Cf. A377893.

nmax=17; CoefficientList[(1/x)InverseSeries[Series[x*Exp[-3*x]/(1 + x)^3 ,{x,0,nmax}]],x]Range[0,nmax-1]! (* Stefano Spezia, Feb 06 2025 *)

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

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^3 * exp(3 * x * A(x)).
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A377893.
a(n) = 3 * n! * Sum_{k=0..n} (3*n+3)^(k-1) * binomial(3*n+3,n-k)/k!.

A380826 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3x) / (1 + xexp(-2*x)) ).

Original entry on oeis.org

1, 4, 43, 810, 22273, 811728, 36979467, 2025462736, 129748802401, 9522843081984, 788169731306059, 72641846664240384, 7379343546762675873, 819269203286474309632, 98698960328223628470379, 12824232015954542746048512, 1787731339345567827140060737, 266157254062414638948185210880

Offset: 0

Seiichi Manyama, Feb 04 2025

nonn

Index entries for reversions of series

Cf. A088690, A161633, A380808.

Cf. A361182, A380829, A380830.

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

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

a(n) = n! * Sum_{k=0..n} (n+2*k+3)^k * binomial(n,k)/(k+1)!.

A380828 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x) / (1 + x) ).

Original entry on oeis.org

1, 3, 26, 398, 8904, 264072, 9790192, 436382256, 22748241024, 1358633214080, 91503397265664, 6862436244211968, 567252637423922176, 51244493078278198272, 5023312927780022323200, 531082672018567209801728, 60239691905397303186849792, 7297357396264290237329473536

Offset: 0

Seiichi Manyama, Feb 05 2025

nonn

Index entries for reversions of series

Cf. A352448, A380808.
Cf. A088690, A380830.
Cf. A376093.

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

E.g.f. A(x) satisfies A(x) = exp(2*x*A(x)) / ( 1 - x*exp(2*x*A(x)) ).
a(n) = n! * Sum_{k=0..n} (2*n+2)^k * binomial(n,k)/(k+1)!.
a(n) = A376093(n+1)/(n+1).

A380830 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x) / (1 + x) ).

Original entry on oeis.org

1, 4, 47, 978, 29769, 1201728, 60656679, 3681441648, 261337079601, 21256149703680, 1949700750690879, 199146039242552064, 22420399033075845177, 2758645779752490872832, 368321963942753147683575, 53038788218443786432223232, 8194316429830951008255159009, 1352065789150879084276947222528

Offset: 0

Seiichi Manyama, Feb 05 2025

nonn

Index entries for reversions of series

Cf. A361182, A380826, A380829.
Cf. A088690, A380828.
Cf. A376094.

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

E.g.f. A(x) satisfies A(x) = exp(3*x*A(x)) / ( 1 - x*exp(3*x*A(x)) ).
a(n) = n! * Sum_{k=0..n} (3*n+3)^k * binomial(n,k)/(k+1)!.
a(n) = A376094(n+1)/(n+1).

A088693 E.g.f: A(x) = f(xA(x)^2), where f(x) = (1+3x)*exp(x).

Original entry on oeis.org

1, 4, 71, 2434, 126117, 8804776, 775425427, 82565249670, 10319537275913, 1481520436347628, 240291243489544191, 43458295155840595306, 8672066947756086825325, 1892794863486905965709136, 448582856421716543783775947, 114720816495997657177701763246

Offset: 0

Paul D. Hanna, Oct 07 2003

nonn

Radius of convergence of A(x): r = (2/27)*exp(-1/3) = 0.053076..., where A(r) = (3/2)*exp(1/6) and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.

Cf. A088690, A088692.

Table[n!*SeriesCoefficient[((1+3*x)*E^x)^(2*n+1)/(2*n+1),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 24 2014 *)

a(n)=n!*polcoeff(((1+3*x)*exp(x))^(2*n+1)+x*O(x^n),n,x)/(2*n+1)

a(n) = n! * [x^n] ((1+3*x)*exp(x))^(2*n+1)/(2*n+1).

a(n) ~ 3^(3*n+2) * n^(n-1) / (sqrt(7) * 2^(n+2) * exp(2*n/3-1/6)). - Vaclav Kotesovec, Jan 24 2014

A379846 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + xexp(2x)) ).

Original entry on oeis.org

1, 2, 15, 202, 3993, 104896, 3449431, 136490768, 6319722513, 335372124160, 20074806151551, 1338341234648320, 98356732036224745, 7900673166769620992, 688709957632464564231, 64754459774124307019776, 6532479591772426224737697, 703834470938326183482621952

Offset: 0

Seiichi Manyama, Jan 04 2025

nonn

Cf. A088690, A379456, A379847.

Cf. A366232, A379701.

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

a(n) = (n!/(n+1)) * Sum_{k=0..n} (3*n-2*k+1)^k * binomial(n+1,n-k)/k!.

E.g.f. A(x) satisfies A(x) = exp(x*A(x)) / ( 1 - x*exp(3*x*A(x)) ). - Seiichi Manyama, Feb 04 2025

A379847 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + xexp(3x)) ).

Original entry on oeis.org

1, 2, 17, 259, 5773, 171021, 6342937, 283094309, 14785425081, 885090944809, 59765476266061, 4494836808752049, 372655043070926821, 33769844474642217293, 3320996349535681398849, 352267766021524028011981, 40091829710459334010532593, 4873329774181782935197522641

Offset: 0

Seiichi Manyama, Jan 04 2025

nonn

Cf. A088690, A379456, A379846.

Cf. A366233, A379702.

a(n) = n!*sum(k=0, n, (4*n-3*k+1)^k*binomial(n+1, n-k)/k!)/(n+1);

a(n) = (n!/(n+1)) * Sum_{k=0..n} (4*n-3*k+1)^k * binomial(n+1,n-k)/k!.

E.g.f. A(x) satisfies A(x) = exp(x*A(x)) / ( 1 - x*exp(4*x*A(x)) ). - Seiichi Manyama, Feb 04 2025

A379897 E.g.f. A(x) satisfies A(x) = 1/(exp(-x*A(x)^3) - x).

Original entry on oeis.org

1, 2, 19, 388, 12273, 528216, 28824811, 1907463440, 148449329825, 13287501321472, 1344889039128291, 151888157696186880, 18936317798871433681, 2583256803370493809664, 382764484828432552194875, 61215815097927618654693376, 10510472883169375744953509697, 1928296235410784800904193638400

Offset: 0

Seiichi Manyama, Jan 05 2025

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..319

Cf. A377893, A379870, A379912.

Cf. A088690, A379884.

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

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

cp's OEIS Frontend

A379936 E.g.f. A(x) satisfies A(x) = 1/( exp(-x*A(x)^(1/2)) - x )^2.

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A380646 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x)/(1 + x)^2 ).

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A380647 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x)/(1 + x)^3 ).

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A380826 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x) / (1 + x*exp(-2*x)) ).

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A380828 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x) / (1 + x) ).

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A380830 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x) / (1 + x) ).

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A088693 E.g.f: A(x) = f(x*A(x)^2), where f(x) = (1+3*x)*exp(x).

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A379846 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + x*exp(2*x)) ).

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A379847 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + x*exp(3*x)) ).

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A380826 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3x) / (1 + xexp(-2*x)) ).

A088693 E.g.f: A(x) = f(xA(x)^2), where f(x) = (1+3x)*exp(x).

A379846 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + xexp(2x)) ).

A379847 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + xexp(3x)) ).