cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A049214 Scaled coefficients of (arctanh x)^3.

Original entry on oeis.org

1, 20, 784, 52352, 5360256, 782525952, 154594381824, 39746508226560, 12902483299368960, 5162443736924160000, 2496471943395999744000, 1435556519572510605312000, 968234590214616380866560000
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Examples

			(arctanh x)^3 = x^3 + x^5 + 14/15*x^7 + 818/945*x^9 + ...

Crossrefs

Cf. A049215, A049216, A049217.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[ArcTanh[x]^3, {x, 0, 2n+3}]*(2n+3)!/6;
a /@ Range[0, 12] (* Jean-François Alcover, Mar 26 2021 *)

Formula

E.g.f.: (arctanh x)^3 or (1-x^2)^-1 * (arctanh x)^2.
a(n) is coefficient of x^(2*n+3) in (arctanh x)^3, multiplied by (2*n+3)!/6.
a(n) = (1/4)* Sum_{p=1..2*n+2} binomial(2*n+1,p-1) Sum_{m=3..2*n+3} binomial(m,3)*abs(s(2*n+3,m))*((-p)^(m-3)+(-p)^(m-3)) where s(n,m) is Stirling number of first kind . - Ammar Khatab, Aug 13 2024

A049215 Scaled coefficients of (arctanh x)^4.

Original entry on oeis.org

1, 40, 2464, 229760, 30633856, 5561407488, 1322489954304, 399463775797248, 149519094622027776, 67960463478175825920, 36884466463352967659520, 23566236397584291201024000, 17511732541318788803985408000
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Examples

			(arctanh x)^4 = x^4 + 4/3*x^6 + 22/15*x^8 + 1436/945*x^10 + ...

Crossrefs

Cf. A049214, A049216, A049217.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[ArcTanh[x]^4, {x, 0, 2n+4}]*(2n+4)!/4!;
a /@ Range[0, 12] (* Jean-François Alcover, Mar 26 2021 *)

Formula

E.g.f.: (arctanh x)^4 or (1-x^2)^-1 * (arctanh x)^3.
a(n) is coefficient of x^(2*n+4) in (arctanh x)^4, multiplied by (2*n+4)!/4!.
a(n) = (1/8) * Sum_{p=1..2*n+3} binomial(2*n+2,p-1)*Sum_{m=4..2*n+4} binomial(m,4)*abs(s(2*n+4,m))*((-p)^(m-4)+(-p)^(m-4)) where s(n,m) is Stirling number of first kind . - Ammar Khatab, Aug 13 2024

A008309 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.

Original entry on oeis.org

1, 1, -2, 1, -8, 1, 24, -20, 1, 184, -40, 1, -720, 784, -70, 1, -8448, 2464, -112, 1, 40320, -52352, 6384, -168, 1, 648576, -229760, 14448, -240, 1, -3628800, 5360256, -804320, 29568, -330, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			With the zero coefficients included the data begins 1; 0,1; -2,0,1; 0,-8,0,1; 24,0,-20,0,1; 0,184,0,-40,0,1; ..., which is A049218.
The table without zeros begins
    1;
    1;
   -2,   1;
   -8,   1;
   24, -20,   1;
  184, -40,   1;
  ...

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.

Crossrefs

Essentially same as A049218.
A007290(n) = -T(n, floor(n-1)/2);
A010050(n) = (-1)^n*T(2n+1, 1);
A049034(n) = (-1)^n*T(2n+2, 1);
A049214(n) = (-1)^n*T(2n+3, 2);
A049215(n) = (-1)^n*T(2n+4, 2);
A049216(n) = (-1)^n*T(2n+5, 3);
A049217(n) = (-1)^n*T(2n+6, 3).

Programs

  • Mathematica
    t[n_, k_] := (-1)^((3*n+k)/2)*n!/2^k*Sum[2^i*Binomial[n-1, i-1]*StirlingS1[i, k]/i!, {i, k, n}]; Flatten[Table[t[n,k], {n,1,11}, {k, 2-Mod[n, 2], n, 2}]] (* Jean-François Alcover, Aug 31 2011, after Vladimir Kruchinin *)
  • PARI
    a(n)=atan(x)^n/n!
T(n,k)=polcoeff(serlaplace(a(2*k-n%2)), n)

Formula

E.g.f.: arctan(x)^k/k! = Sum_{n>=0} T(m, floor((k+1)/2))* x^m/m!, where m = 2*n + k mod 2.

Extensions

Additional comments from Michael Somos
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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