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A059419
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Triangle T(n,k) (1<=k<=n) of tangent numbers, read by rows: T(n,k) = coefficient of x^n/n! in expansion of (tan x)^k/k!.
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12
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1, 0, 1, 2, 0, 1, 0, 8, 0, 1, 16, 0, 20, 0, 1, 0, 136, 0, 40, 0, 1, 272, 0, 616, 0, 70, 0, 1, 0, 3968, 0, 2016, 0, 112, 0, 1, 7936, 0, 28160, 0, 5376, 0, 168, 0, 1, 0, 176896, 0, 135680, 0, 12432, 0, 240, 0, 1, 353792, 0, 1805056, 0, 508640, 0, 25872, 0, 330, 0, 1, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.
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LINKS
| Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565
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FORMULA
| T(n+1, k) = T(n, k-1) + k*(k+1)*T(n, k+1), T(n, n) = 1.
If n+k is odd, T(n,k) = 0 = (-1)^((n+k)/2)*sum{j=k..n} (j!/n!) *stirling2(n,j) *2^(n-j) *(-1)^(n+j-k) *binomial(j-1,k-1). [From Vladimir Kruchinin, Feb 10 2011]
E.g.f.: exp(t*tan(x))-1 = t*x+t^2*x^2/2!+(2*t+t^3)*x^3/3!+....
The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x^2)*d/dx. - Peter Bala, Nov 25 2011
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EXAMPLE
| 1;
0,1;
2,0,1;
0,8,0,1;
16,0,20,0,1;
0,136,0,40,0,1;
272,0,616,0,70,0,1;
0,3968,0,2016,0,112,0,1;
7936,0,28160,0,5376,0,168,0,1;
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MAPLE
| A059419 := proc(n, k) option remember; if n = k then 1; elif k <0 or k > n then 0; else procname(n-1, k-1)+k*(k+1)*procname(n-1, k+1) ; end if; end proc: # R. J. Mathar, Feb 11 2011
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MATHEMATICA
| d[f_ ] := (1+x^2)*D[f, x]; d[ f_, n_] := Nest[d, f, n]; row[n_] := Rest[ CoefficientList[ d[Exp[x*t], n] /. x -> 0, t]]; Flatten[ Table[ row[n], {n, 1, 12}]] (* From Jean-François Alcover, Dec 21 2011, after Peter Bala *)
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PROG
| (PARI) T(n, k)=if(k<1|k>n, 0, n!*polcoeff(tan(x+x*O(x^n))^k/k!, n))
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CROSSREFS
| Diagonals give A000182, A024283, A059420 (interspersed with 0's), also A007290, A059421. Row sums give A006229. Essentially the same triangle as A008308.
A111593 (signed triangle with extra column k=0 and row n=0).
Sequence in context: A095403 A011328 A048277 * A185415 A049218 A154469
Adjacent sequences: A059416 A059417 A059418 * A059420 A059421 A059422
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KEYWORD
| nonn,easy,nice,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2001
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Feb 01 2001
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