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

%I A008309 #31 Jun 29 2025 20:19:25
%S A008309 1,1,-2,1,-8,1,24,-20,1,184,-40,1,-720,784,-70,1,-8448,2464,-112,1,
%T A008309 40320,-52352,6384,-168,1,648576,-229760,14448,-240,1,-3628800,
%U A008309 5360256,-804320,29568,-330,1
%N A008309 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.
%D A008309 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.
%F A008309 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.
%e A008309 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.
%e A008309 The table without zeros begins
%e A008309     1;
%e A008309     1;
%e A008309    -2,   1;
%e A008309    -8,   1;
%e A008309    24, -20,   1;
%e A008309   184, -40,   1;
%e A008309   ...
%t A008309 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_ *)
%o A008309 (PARI)
%o A008309 a(n)=atan(x)^n/n!
%o A008309 T(n,k)=polcoeff(serlaplace(a(2*k-n%2)), n)
%Y A008309 Essentially same as A049218.
%Y A008309 A007290(n) = -T(n, floor(n-1)/2);
%Y A008309 A010050(n) = (-1)^n*T(2n+1, 1);
%Y A008309 A049034(n) = (-1)^n*T(2n+2, 1);
%Y A008309 A049214(n) = (-1)^n*T(2n+3, 2);
%Y A008309 A049215(n) = (-1)^n*T(2n+4, 2);
%Y A008309 A049216(n) = (-1)^n*T(2n+5, 3);
%Y A008309 A049217(n) = (-1)^n*T(2n+6, 3).
%K A008309 sign,tabf,nice
%O A008309 1,3
%A A008309 _N. J. A. Sloane_
%E A008309 Additional comments from _Michael Somos_