A007289 -id:A007289 - cp's OEIS frontend

A385306 Expansion of e.g.f. 1/(1 - 2 * sin(x))^(1/2).

1, 1, 3, 14, 93, 796, 8343, 103424, 1479993, 24008656, 435364683, 8726775584, 191601310293, 4572794295616, 117871476051423, 3263515787807744, 96591500816346993, 3043368045293138176, 101702692426476460563, 3592948632452749243904, 133794496537591022166093

Offset: 0

Seiichi Manyama, Jun 24 2025

nonn

Cf. A000111, A385307.
Cf. A380015, A385304, A385308, A385310.
Cf. A001147, A001586, A007289, A136630.

With[{nn=20},CoefficientList[Series[1/Sqrt[1-2Sin[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 09 2025 *)

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*I^(n-k)*a136630(n, k));

a(n) = Sum_{k=0..n} A001147(k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

a(n) ~ 2^(n+1) * 3^(n + 1/4) * n^n / (exp(n) * Pi^(n + 1/2)). - Vaclav Kotesovec, Jun 28 2025

A352638 Expansion of e.g.f. 1/(1 - 3*sin(x)).

Original entry on oeis.org

1, 3, 18, 159, 1872, 27543, 486288, 10016619, 235798272, 6244714443, 183756215808, 5947907121879, 210026879004672, 8034293365747743, 330982609573398528, 14609181655918083939, 687820834029346947072, 34407546247054875367443, 1822450167175258689896448

Offset: 0

Seiichi Manyama, Mar 25 2022

nonn

Cf. A000111, A007289.
Cf. A107403, A352640.
Cf. A136630.

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

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

a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (-1)^k*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 * binomial(n,2*k+1) * a(n-2*k-1).
a(n) ~ n! / (2^(3/2) * arcsin(1/3)^(n+1)). - Vaclav Kotesovec, Mar 26 2022
a(n) = Sum_{k=0..n} 3^k * k! * i^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 25 2025

A193474 Table read by rows: The coefficients of the polynomials P(n, x) = Sum{k=0..n} Sum{j=0..k} (-1)^j * 2^(-k) * binomial(k, j) * (k-2j)^n x^(n-k).

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 6, 0, 1, 0, 24, 0, 8, 0, 0, 120, 0, 60, 0, 1, 0, 720, 0, 480, 0, 32, 0, 0, 5040, 0, 4200, 0, 546, 0, 1, 0, 40320, 0, 40320, 0, 8064, 0, 128, 0, 0, 362880, 0, 423360, 0, 115920, 0, 4920, 0, 1, 0, 3628800, 0, 4838400, 0, 1693440, 0, 130560, 0, 512, 0, 0

Offset: 1

Peter Luschny, Aug 01 2011

See A196776 for a row reversed form of this triangle. - Peter Bala, Oct 06 2011

			The sequence of polynomials P(n, x) begins:
[0]    1;
[1]    1;
[2]    2;
[3]    6 +      x^2;
[4]   24 +    8*x^2;
[5]  120 +   60*x^2 +     x^4;
[6]  720 +  480*x^2 +  32*x^4;
[7] 5040 + 4200*x^2 + 546*x^4 + x^6.

Cf. A196776, A000142, A006154, A191277, A000111, A000828, A000557, A107403, A007289, A005990.

A193474_polynom := proc(n,x) local k, j;
add(add((-1)^j*2^(-k)*binomial(k,j)*(k-2*j)^n*x^(n-k),j=0..k),k=0..n) end: seq(seq(coeff(A193474_polynom(n,x),x,i),i=0..n),n=0..10);

p[n_, x_] := Sum[(-1)^j*2^(-k)*Binomial[k, j]*(k-2*j)^n*x^(n-k), {k, 0, n}, {j, 0, k}]; t[n_, k_] := Coefficient[p[n, x], x, k]; t[0, 0] = 1; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2014 *)

P(n, 0) = A000142(n).
P(n, 1) = A006154(n).
P(n, 2) = A191277(n).
P(n, i) = A000111(n+1), where i is the imaginary unit.
P(n, i)*2^n = A000828(n+1).
P(n, 1/2)*2^n = A000557(n).
P(n, 1/3)*3^n = A107403(n).
P(n, i/2)*2^n = A007289(n).
G(m, x) = 1/(1 - m*sinh(x)) is the generating function of m^n*P(n, 1/m).
GI(m, x) = 1/(1 - m*sin(x)) is the generating function of m^n*P(n, i/m).
[x^2] P(n+1, x) = A005990(n).

A385367 Expansion of e.g.f. 1/(1 - 2 * arcsinh(x)).

Original entry on oeis.org

1, 2, 8, 46, 352, 3378, 38912, 522702, 8024064, 138586722, 2659565568, 56141737518, 1292851544064, 32253357421842, 866534937329664, 24943658876605902, 765883864848531456, 24985882009464388290, 863077992845681885184, 31469256501815056673070

Offset: 0

Seiichi Manyama, Jun 26 2025

Cf. A296675, A385368.

Cf. A007289, A385343, A385346, A385371.

With[{nn=20},CoefficientList[Series[1/(1-2ArcSinh[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 14 2025 *)

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

E.g.f.: 1/(1 - 2 * log(x + sqrt(x^2 + 1))).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A385371.
a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ sqrt(Pi) * (1 + exp(1)) * 2^(n - 1/2) * n^(n + 1/2) / ((exp(1) - 1)^(n+1) * exp(n/2)). - Vaclav Kotesovec, Jun 27 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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A385306 Expansion of e.g.f. 1/(1 - 2 * sin(x))^(1/2).

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A352638 Expansion of e.g.f. 1/(1 - 3*sin(x)).

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A193474 Table read by rows: The coefficients of the polynomials P(n, x) = Sum{k=0..n} Sum{j=0..k} (-1)^j * 2^(-k) * binomial(k, j) * (k-2*j)^n * x^(n-k).

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A385367 Expansion of e.g.f. 1/(1 - 2 * arcsinh(x)).

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

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A193474 Table read by rows: The coefficients of the polynomials P(n, x) = Sum{k=0..n} Sum{j=0..k} (-1)^j * 2^(-k) * binomial(k, j) * (k-2j)^n x^(n-k).