A194471 -id:A194471 - cp's OEIS frontend

A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4xexp(x))).

1, 1, 6, 57, 796, 14785, 344046, 9640225, 316255416, 11896233345, 504918768250, 23874754106401, 1244712973780068, 70940791877082049, 4388291507415513894, 292823509879910802465, 20966854494419642792176, 1603540841320336494905089, 130464295561360336835272050

Offset: 0

Ilya Gutkovskiy, Nov 18 2017

nonn

Inverse binomial transform of A194471.

Seiichi Manyama, Table of n, a(n) for n = 0..356

Cf. A000108, A006531, A052895, A194471, A213644, A295239.

a:=series(2/(1+sqrt(1-4*x*exp(x))),x=0,19): seq(n!*coeff(a,x,n),n=0..18); # Paolo P. Lava, Mar 27 2019

nmax = 18; CoefficientList[Series[2/(1 + Sqrt[1 - 4 x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 18; CoefficientList[Series[1/(1 + ContinuedFractionK[-x Exp[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n, k] k! Sum[(m + 1)^(k - m - 1) Binomial[2 m, m]/(k - m)!, {m, 0, k}], {k, 0, n}], {n, 0, 18}]

a(n) = n!*sum(k=0, n, k^(n-k)*binomial(2*k, k)/((k+1)*(n-k)!)); \\ Seiichi Manyama, Aug 15 2023

E.g.f.: 1/(1 - x*exp(x)/(1 - x*exp(x)/(1 - x*exp(x)/(1 - x*exp(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2*(1 + LambertW(1/4))) * n^(n-1) / ((LambertW(1/4))^n * exp(n)). - Vaclav Kotesovec, Nov 18 2017
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*k+1,k)/( (2*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A000108(k)/(n-k)!. - Seiichi Manyama, Aug 15 2023

A377742 E.g.f. satisfies A(x) = exp(x) / (1 - x * A(x))^2.

Original entry on oeis.org

1, 3, 23, 331, 7133, 205901, 7470475, 326932299, 16768124217, 986753701657, 65548017270791, 4852285640543639, 396133183892522389, 35359325061987638661, 3426053898460864501251, 358128187005971803014211, 40172982580368589391407217, 4813677071886578522596221233

Offset: 0

Seiichi Manyama, Nov 05 2024

nonn

Cf. A194471, A377743, A377744.
Cf. A001339, A377745.
Cf. A377503.

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

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

A371318 E.g.f. satisfies A(x) = exp(x) + x*A(x)^3.

Original entry on oeis.org

1, 2, 13, 190, 4345, 135346, 5345749, 256004974, 14416470961, 933597699202, 68358972056221, 5584583237569150, 503607231488672425, 49690178089937051122, 5325031693664693833957, 615922452708451717999726, 76479190243720703567763553

Offset: 0

Seiichi Manyama, Mar 18 2024

nonn

Cf. A000522, A194471, A373324.

Table[n! Sum[(2 k + 1)^(n - k - 1)*Binomial[3 k, k]/(n - k)!, {k, 0, n}], {n, 0, 20}] (* Wesley Ivan Hurt, May 25 2024 *)

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

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

a(n) ~ sqrt(1 + LambertW(8/27)) * 2^n * n^(n-1) / (3 * exp(n) * LambertW(8/27)^(n + 1/2)). - Vaclav Kotesovec, Jun 01 2024

A373324 E.g.f. satisfies A(x) = exp(x) + x*A(x)^4.

Original entry on oeis.org

1, 2, 17, 349, 11249, 495401, 27715225, 1882157369, 150415131809, 13830661215649, 1438437863857961, 166962406866895817, 21396540301232809201, 3000661115664455591921, 457109095827413086174265, 75165845570197217863619161, 13270031366484750565975875905, 2503433069466253671859276038977

Offset: 0

Vaclav Kotesovec, Jun 01 2024

nonn

In general, for k > 1, if e.g.f. satisfies A(x) = exp(x) + x*A(x)^k, then a(n) ~ sqrt(1 + LambertW((1 - 1/k)^k)) * (k-1)^(n - 1/2 + 1/(k-1)) * n^(n-1) / (k^(1/2 + 1/(k-1)) * exp(n) * LambertW((1 - 1/k)^k)^(n + 1/(k-1))).

Cf. A000522, A194471, A371318.

Table[n! * Sum[(3*k+1)^(n-k-1) * Binomial[4*k,k] / (n-k)!, {k, 0, n}], {n, 0, 20}]

a(n) ~ sqrt(1 + LambertW(81/256)) * 3^(n - 1/6) * n^(n-1) / (2^(5/3) * exp(n) * LambertW(81/256)^(n + 1/3)).

A377743 E.g.f. satisfies A(x) = exp(x) / (1 - x * A(x))^3.

Original entry on oeis.org

1, 4, 43, 853, 25141, 989581, 48885187, 2910389875, 202958554057, 16233163690537, 1465257396236551, 147359765665925143, 16341437664329027389, 1981169884084699982701, 260701144663332062732491, 37007345616327485166160651, 5637148375602304430334748945, 917186940500490837457393476817

Offset: 0

Seiichi Manyama, Nov 06 2024

nonn

Cf. A194471, A377742, A377744.

Cf. A373324, A377504.

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

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k-1) * binomial(4*k+2,k)/(n-k)!.

A194958 E.g.f. satisfies: A(x) = cosh(x) + x*A(x)^2.

Original entry on oeis.org

1, 1, 5, 36, 409, 6280, 121501, 2839424, 77834737, 2449360512, 87040440181, 3447798906112, 150645874207753, 7197909122453504, 373365727806824845, 20895734364795187200, 1255062315134651501281, 80528111291313595580416, 5497183726333878664852453

Offset: 0

Paul D. Hanna, Sep 06 2011

nonn

The radius of convergence r of the e.g.f. A(x) satisfies: r = 1/(4*cosh(r)) = limit (n+1)*a(n)/a(n+1) = 0.24280736240... with A(r) = 1/(2*r) = 2.059245630...

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 36*x^3/3! + 409*x^4/4! + 6280*x^5/5! +...
Related expansion:
A(x)^2 = 1 + 2*x + 12*x^2/2! + 102*x^3/3! + 1256*x^4/4! + 20250*x^5/5! +...

G. C. Greubel, Table of n, a(n) for n = 0..360

Cf. A194957, A194471.

CoefficientList[Series[(1-Sqrt[1-4*x*Cosh[x]])/(2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 21 2013 *)

{a(n)=n!*polcoeff((1 - sqrt(1 - 4*x*cosh(x +O(x^(n+2))))) / (2*x),n)}

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

E.g.f.: A(x) = (1 - sqrt(1 - 4*x*cosh(x))) / (2*x).
a(n) = (1 + (-1)^n)/2 + n*Sum_{k=0..n-1} C(n-1,k)*a(k)*a(n-1-k) for n>=0.
a(n) ~ n! * sqrt(cosh(r)+r*sinh(r))/(2*r^(n+1/2)*sqrt(Pi)*n^(3/2)), where r = 0.2428073624... is defined in the comment. - Vaclav Kotesovec, Sep 21 2013

A194957 E.g.f. satisfies: A(x) = cos(x) + x*A(x)^2.

Original entry on oeis.org

1, 1, 3, 24, 265, 3880, 71099, 1568896, 40538065, 1201214592, 40167734579, 1496537977088, 61482321148249, 2761475028475904, 134626530031186283, 7080317004476928000, 399585298475488156321, 24087998749960900673536, 1544790381323645227703651

Offset: 0

Paul D. Hanna, Sep 06 2011

nonn

The radius of convergence r of the e.g.f. A(x) satisfies: r = 1/(4*cos(r)) = limit (n+1)*a(n)/a(n+1) = 0.2585985822541... with A(r) = 1/(2*r) = 1.933498612565961...

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 24*x^3/3! + 265*x^4/4! + 3880*x^5/5! + ...
Related expansion:
A(x)^2 = 1 + 2*x + 8*x^2/2! + 66*x^3/3! + 776*x^4/4! + 11850*x^5/5! + ...

G. C. Greubel, Table of n, a(n) for n = 0..365

Cf. A194958, A194471.

CoefficientList[Series[(1-Sqrt[1-4*x*Cos[x]])/(2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 21 2013 *)

{a(n)=n!*polcoeff((1 - sqrt(1 - 4*x*cos(x +O(x^(n+2))))) / (2*x),n)}

{a(n)=(-1)^(n\2)*(1 + (-1)^n)/2 +n*sum(k=0, n-1, binomial(n-1, k)*a(k)*a(n-1-k))}

E.g.f.: A(x) = (1 - sqrt(1 - 4*x*cos(x))) / (2*x).
a(n) = (-1)^floor(n/2)*(1 + (-1)^n)/2 + n*Sum_{k=0..n-1} binomial(n-1,k)*a(k)*a(n-1-k) for n >= 0.
a(n) ~ n! * sqrt(cos(r)-sin(r)*r)/(2*sqrt(Pi)*r^(n+1/2)*n^(3/2)), where r = 0.258598582254189... is defined in the comment. - Vaclav Kotesovec, Sep 21 2013

A377744 E.g.f. satisfies A(x) = exp(x) / (1 - x * A(x))^4.

Original entry on oeis.org

1, 5, 69, 1741, 65025, 3238401, 202252549, 15216086789, 1340493558497, 135418524663457, 15436319894361141, 1960277599669850517, 274474966233168968353, 42012725272366653895169, 6979546631782182590117189, 1250777360824265136694022341, 240516661686854988775792192833

Offset: 0

Seiichi Manyama, Nov 06 2024

nonn

Cf. A194471, A377742, A377743.

Cf. A377528.

a(n) = n!*sum(k=0, n, (k+1)^(n-k-1)*binomial(5*k+3, k)/(n-k)!);

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k-1) * binomial(5*k+3,k)/(n-k)!.

A380717 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2 * (1 - xA(x))^2) / (1 - xA(x)).

Original entry on oeis.org

1, 2, 13, 151, 2561, 57401, 1602985, 53659453, 2095244289, 93523526065, 4698386208521, 262397580544133, 16128832249562785, 1082120615743840297, 78695060375718726633, 6166431270471329586301, 517970728078392717716225, 46432097598077316120950369, 4424506354750061857673476873

Offset: 0

Seiichi Manyama, Jan 30 2025

nonn

Cf. A194471, A380673.

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

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

cp's OEIS Frontend

A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*exp(x))).

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A377742 E.g.f. satisfies A(x) = exp(x) / (1 - x * A(x))^2.

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A371318 E.g.f. satisfies A(x) = exp(x) + x*A(x)^3.

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A373324 E.g.f. satisfies A(x) = exp(x) + x*A(x)^4.

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A377743 E.g.f. satisfies A(x) = exp(x) / (1 - x * A(x))^3.

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A194958 E.g.f. satisfies: A(x) = cosh(x) + x*A(x)^2.

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A194957 E.g.f. satisfies: A(x) = cos(x) + x*A(x)^2.

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A377744 E.g.f. satisfies A(x) = exp(x) / (1 - x * A(x))^4.

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A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4xexp(x))).

A380717 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2 * (1 - xA(x))^2) / (1 - xA(x)).