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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
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OFFSET
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1,3
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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;
...
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MATHEMATICA
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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 *)
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PROG
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(PARI) T(n, k)=polcoeff(serlaplace(a(2*k-n%2)), n) where a(n)=atan(x)^n/n!
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CROSSREFS
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KEYWORD
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sign,tabf,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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