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

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

Author

N. J. A. Sloane

Keywords

• sign
• tabf
• nice

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