A352642 -id:A352642 - cp's OEIS frontend

A352643 Expansion of e.g.f. exp(3 * x * cos(x)).

1, 3, 9, 18, -27, -552, -3051, -3504, 102825, 1043712, 3192129, -41548416, -653192883, -3033406464, 31367500173, 670266381312, 3916411302609, -40460110970880, -1038593550985479, -6810646726410240, 82445831323038261, 2280185182260854784, 15300402721484153733

Offset: 0

Seiichi Manyama, Mar 25 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..536

Cf. A009189, A185951, A352642.

With[{m = 22}, Range[0, m]! * CoefficientList[Series[Exp[3*x*Cos[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(3*x*cos(x))))

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

a(n) = Sum_{k=0..n} 3^k * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 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

A352639 Expansion of e.g.f. exp(2*sin(x)).

Original entry on oeis.org

1, 2, 4, 6, 0, -46, -192, -266, 1792, 14114, 34816, -171930, -2027520, -6522382, 34750464, 496296022, 1748500480, -12731696062, -186550845440, -617309234490, 7292215885824, 99199654760978, 248883934396416, -5836506132182090, -69729013345550336

Offset: 0

Seiichi Manyama, Mar 25 2022

sign

Cf. A002017, A136630, A352640.

Cf. A007289, A352279, A352642.

With[{m = 24}, Range[0, m]! * CoefficientList[Series[Exp[2*Sin[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sin(x))))

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

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

a(n) = Sum_{k=0..n} 2^k * i^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 2025

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A352643 Expansion of e.g.f. exp(3 * x * cos(x)).

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

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A352639 Expansion of e.g.f. exp(2*sin(x)).

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