A185951 -id:A185951 - cp's OEIS frontend

A385309 Expansion of e.g.f. 1/(1 - 3 * x * cosh(x))^(1/3).

1, 1, 4, 31, 328, 4485, 75520, 1509347, 34916224, 917703145, 27011107840, 880133628231, 31451749424128, 1223047891889837, 51414400611438592, 2323391075748100555, 112315439676217262080, 5783449255108473820497, 316034972288791445241856, 18265740423344520141491951

Offset: 0

Seiichi Manyama, Jun 24 2025

nonn

Cf. A205571, A385308.
Cf. A380017, A385305, A385307, A385311.
Cf. A007559, A185951, A352649, A385282.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*a185951(n, k));

a(n) = Sum_{k=0..n} A007559(k) * A185951(n,k), where A185951(n,0) = 0^n.

A215364 E.g.f. A(x) satisfies A(x) = 1 + xA(x)^2cosh(x*A(x)).

Original entry on oeis.org

1, 1, 4, 33, 408, 6725, 139200, 3475717, 101722880, 3416079753, 129507425280, 5471712276041, 254965505507328, 12990483544072333, 718474796305989632, 42871067358096134445, 2745230569464318197760, 187780115708775158008337, 13665196427126843296972800

Offset: 0

Paul D. Hanna, Aug 08 2012

nonn

			E.g.f: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 408*x^4/4! + 6725*x^5/5! +...

Cf. A162654, A185951, A215363, A213644.

CoefficientList[1/x*InverseSeries[Series[x-x^2*Cosh[x], {x, 0, 21}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 13 2014 *)

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

{a(n)=n!*polcoeff((1/x)*serreverse(x-x^2*cosh(x+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n+k+1,k)/(n+k+1)*a185951(n, k)); \\ Seiichi Manyama, Feb 16 2025

E.g.f. satisfies:
(1) A(x) = (1/x)*Series_Reversion(x-x^2*cosh(x)).
(2) A(x) = 1/(1 - x*A(x)*cosh(x*A(x))).
(3) A(x-x^2*cosh(x)) = 1/(1-x*cosh(x)).
a(n) = [x^n/n!] 1/(1 - x*cosh(x))^(n+1) / (n+1).
a(n) ~ n^(n-1) * s*sqrt(1/(6-2*s-r^2*s^2+r^2*s^3)) / (exp(n) * r^n), where r = 0.2278231894714399793... and s = 1.855593992316816009... are the roots of the equations r*s*(2*cosh(r*s) + r*s*sinh(r*s)) = 1, 1 + r*s^2*cosh(r*s) = s. - Vaclav Kotesovec, Jan 13 2014
a(n) = Sum_{k=0..n} k! * binomial(n+k+1,k)/(n+k+1) * A185951(n,k). - Seiichi Manyama, Feb 16 2025

A352646 Expansion of e.g.f. 1/(1 - 2 * x * cos(x)).

Original entry on oeis.org

1, 2, 8, 42, 288, 2410, 24000, 277186, 3648512, 53936082, 885150720, 15970846298, 314273439744, 6698574264122, 153746319720448, 3780677636321010, 99163499845386240, 2763481838977368994, 81542013760903053312, 2539717324111483027594

Offset: 0

Seiichi Manyama, Mar 25 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..415

Cf. A352252, A352647.
Cf. A352642, A352648.
Cf. A185951.

With[{m = 19}, Range[0, m]! * CoefficientList[Series[1/(1 - 2*x*Cos[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

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

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

a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) ~ n! / ((1 - r * sqrt(4*r^2 - 1)) * r^n), where r = A196603 = 0.6100312844641759753709630735134103246737209791121692378637516075328... is the root of the equation 2*r*cos(r) = 1. - Vaclav Kotesovec, Mar 27 2022
a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 25 2025

A352647 Expansion of e.g.f. 1/(1 - 3 * x * cos(x)).

Original entry on oeis.org

1, 3, 18, 153, 1728, 24315, 410400, 8079729, 181786752, 4601232243, 129402385920, 4003157532297, 135098815002624, 4939266681129963, 194472450526169088, 8203835046344538465, 369151362125290045440, 17649035213360472293091, 893431062200523039178752

Offset: 0

Seiichi Manyama, Mar 25 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..384

Cf. A352252, A352646.
Cf. A352643, A352649.
Cf. A185951.

With[{m = 18}, Range[0, m]! * CoefficientList[Series[1/(1 - 3*x*Cos[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x*cos(x))))

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

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

a(n) = Sum_{k=0..n} 3^k * k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 25 2025

A352648 Expansion of e.g.f. 1/(1 - 2 * x * cosh(x)).

Original entry on oeis.org

1, 2, 8, 54, 480, 5290, 70080, 1083614, 19145728, 380552274, 8404669440, 204182993542, 5411361939456, 155365918497530, 4803852288901120, 159142710151610670, 5623576097060290560, 211138456468635968674, 8393550198348236193792, 352212802264773650385110

Offset: 0

Seiichi Manyama, Mar 25 2022

nonn

Cf. A205571, A352649.

Cf. A185951.

With[{m = 19}, Range[0, m]! * CoefficientList[Series[1/(1 - 2*x*Cosh[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

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

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

a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) ~ n! / ((1 + r * sqrt(1 - 4*r^2)) * r^n), where r = 0.452787214835453627588998503316635625709288535855800416726... is the root of the equation 2*r*cosh(r) = 1. - Vaclav Kotesovec, Mar 27 2022
a(n) = Sum_{k=0..n} 2^k * k! * A185951(n,k). - Seiichi Manyama, Jun 25 2025

A352649 Expansion of e.g.f. 1/(1 - 3 * x * cosh(x)).

Original entry on oeis.org

1, 3, 18, 171, 2160, 34035, 643680, 14203371, 358178688, 10161542691, 320315005440, 11106766229163, 420132741912576, 17216605635562515, 759789379494512640, 35925442734363182955, 1811923104577065615360, 97097117111612660889411, 5509300889675218610552832

Offset: 0

Seiichi Manyama, Mar 25 2022

nonn

Cf. A205571, A352648.

Cf. A185951.

With[{m = 17}, Range[0, m]! * CoefficientList[Series[1/(1 - 3*x*Cosh[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x*cosh(x))))

a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (2*k+1)*binomial(n, 2*k+1)*a(n-2*k-1)));

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

a(n) = Sum_{k=0..n} 3^k * k! * A185951(n,k). - Seiichi Manyama, Jun 25 2025

A381141 Expansion of e.g.f. exp( -LambertW(-x * cos(x)) ).

Original entry on oeis.org

1, 1, 3, 13, 89, 821, 9667, 137817, 2306705, 44308009, 960645251, 23205700453, 618086944873, 17996847978461, 568729575572355, 19387150575025201, 709130794848586657, 27704208465508996945, 1151379111946617111043, 50721472225191792506301, 2360928161776701549045241

Offset: 0

Seiichi Manyama, Feb 15 2025

nonn

As stated in the comment of A185951, A185951(n,0) = 0^n.

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A009189, A381144, A381146.

Cf. A185951.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, (k+1)^(k-1)*I^(n-k)*a185951(n, k));

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

a(n) = Sum_{k=0..n} (k+1)^(k-1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381174 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x*cos(x)) ).

Original entry on oeis.org

1, 1, 4, 27, 264, 3365, 52800, 980903, 20984320, 506078505, 13525493760, 394758794419, 12414039171072, 414990179398093, 14523823020621824, 521523225315049215, 18594912994237808640, 613842569215361446097, 14735570097970682265600, -228398321523777856462261

Offset: 0

Seiichi Manyama, Feb 16 2025

sign

As stated in the comment of A185951, A185951(n,0) = 0^n.

Index entries for reversions of series

Cf. A381173, A381175, A381176.

Cf. A185951.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n+k+1, k)/(n+k+1)*I^(n-k)*a185951(n, k));

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

a(n) = Sum_{k=0..n} k! * binomial(n+k+1,k)/(n+k+1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381176 E.g.f. A(x) satisfies A(x) = 1 + xcos(xA(x)).

Original entry on oeis.org

1, 1, 0, -3, -24, -55, 480, 8813, 61824, -264591, -13662720, -185252771, -117427200, 52162650553, 1214778679296, 7998339208845, -370278535495680, -14623177924271263, -202753399336206336, 3863010744775239101, 286065782789626920960, 6603193175290504771881

Offset: 0

Seiichi Manyama, Feb 16 2025

sign

As stated in the comment of A185951, A185951(n,0) = 0^n.

Cf. A381173, A381174, A381175.

Cf. A185951.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n-k+1, k)/(n-k+1)*I^(n-k)*a185951(n, k));

a(n) = Sum_{k=0..n} k! * binomial(n-k+1,k)/(n-k+1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381206 Expansion of e.g.f. 1/(1 - x*cosh(x))^2.

Original entry on oeis.org

1, 2, 6, 30, 192, 1450, 12960, 133574, 1550976, 20055186, 285903360, 4452231982, 75186726912, 1368588922298, 26709799753728, 556339845854550, 12318065768693760, 288894650033594914, 7154212267816648704, 186545064693433665854, 5108590743587243950080

Offset: 0

Seiichi Manyama, Feb 17 2025

nonn

As stated in the comment of A185951, A185951(n,0) = 0^n.

Cf. A205571, A381207.

Cf. A185951.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, (k+1)!*a185951(n, k));

a(n) = Sum_{k=0..n} (k+1)! * A185951(n,k).

cp's OEIS Frontend

A385309 Expansion of e.g.f. 1/(1 - 3 * x * cosh(x))^(1/3).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A215364 E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2*cosh(x*A(x)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A352646 Expansion of e.g.f. 1/(1 - 2 * x * cos(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A352647 Expansion of e.g.f. 1/(1 - 3 * x * cos(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A352648 Expansion of e.g.f. 1/(1 - 2 * x * cosh(x)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A352649 Expansion of e.g.f. 1/(1 - 3 * x * cosh(x)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A381141 Expansion of e.g.f. exp( -LambertW(-x * cos(x)) ).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A381174 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x*cos(x)) ).

Views

Author

Keywords

Comments

Links

Crossrefs

A215364 E.g.f. A(x) satisfies A(x) = 1 + xA(x)^2cosh(x*A(x)).

A381176 E.g.f. A(x) satisfies A(x) = 1 + xcos(xA(x)).