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A008309
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Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.
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2
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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; internal format)
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OFFSET
| 1,3
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.
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FORMULA
| E.g.f.: arctan(x)^k/k!=sum {n=0..inf} T(m, [ k+1 ]/2) x^m/m! where m=2n+k%2.
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EXAMPLE
| 1; 0,1; -2,0,1; 0,-8,0,1; 24,0,-200,0,1; 0,184,0,-40,0,1; ...
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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}]] (* From Jean-François Alcover, Aug 31 2011, after V. Kruchinin *)
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PROG
| (PARI) T(n, k)=polcoeff(serlaplace(a(2*k-n%2)), n) where a(n)=atan(x)^n/n!
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CROSSREFS
| Essentially same as A049218.
A007290(n)=-T(n, [ (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: A176889 A118931 A101280 * A131175 A066532 A205397
Adjacent sequences: A008306 A008307 A008308 * A008310 A008311 A008312
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KEYWORD
| sign,tabf,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Michael Somos.
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