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 A008309 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260. LINKS 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. EXAMPLE 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;   ... 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 *) PROG (PARI) T(n, k)=polcoeff(serlaplace(a(2*k-n%2)), n) where a(n)=atan(x)^n/n! 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). Sequence in context: A118931 A101280 A321280 * A131175 A066532 A205397 Adjacent sequences:  A008306 A008307 A008308 * A008310 A008311 A008312 KEYWORD sign,tabf,nice AUTHOR EXTENSIONS Additional comments from Michael Somos STATUS approved

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Last modified October 14 07:31 EDT 2019. Contains 327995 sequences. (Running on oeis4.)