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A000061 Generalized tangent numbers d(n,1).
(Formerly M0938 N0352)
5
1, 1, 2, 4, 4, 6, 8, 8, 12, 14, 14, 16, 20, 20, 24, 32, 24, 30, 38, 32, 40, 46, 40, 48, 60, 50, 54, 64, 60, 68, 80, 64, 72, 92, 76, 96, 100, 82, 104, 112, 96, 108, 126, 112, 120, 148, 112, 128, 168, 130, 156, 160, 140, 162, 184, 160, 168, 198, 170, 192, 220, 168, 192 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

REFERENCES

Knuth, D. E.; Buckholtz, Thomas J. Computation of tangent, Euler and Bernoulli numbers. Math. Comp. 21 1967 663-688.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Sean A. Irvine, Table of n, a(n) for n = 1..10000

D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 1967 663-688.

D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699

D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Eric Weisstein's World of Mathematics, Tangent Number

FORMULA

From Sean A. Irvine, Mar 26 2012: (Start)

Consider the Dirichlet series L_a(s) = sum_{k>=0} (-a|2k+1) / (2k+1)^s, where (-a|2k+1) is the Jacobi symbol. Then the numbers d(a,n) are defined by L_a(2n)= (pi/(2a))^(2n)*sqrt(a)* d(a,n)/ (2n-1)! for a>1 and n=1,2,3...

From the Shanks paper, these can be computed as:

Write a=bm^2, b squarefree, then d(a,n)=m^(2n-1)(m*prod_(p_i|m)(p_i^(-1))) * prod_(p_i|m)(p_i^(2*n)-jacobi(b,p_i))d(b,n) with d(b,n), b squarefree determined by:

D(b,n)=(-1)^(n-1) * sum(jacobi(k,b)*(b-4k)^(2n-1), k=1..(b-1)/2)  if b == 1(mod 4)

D(b,n)=(-1)^(n-1) * sum(jacobi(b,2k+1)*(b-(2k+1))^(2n-1),2k+1<b) if b != 1(mod 4)

D(b,n)=sum(d(b,n-i)*(-b^2)^i*C(2n-1,2i),i=0..n-1).

D(1,n)=(-1)^(n-1).

Sequence gives a(n)=d(n,1). (End)

CROSSREFS

Cf. A000176.

Sequence in context: A063224 A023847 A279667 * A153176 A229144 A263021

Adjacent sequences:  A000058 A000059 A000060 * A000062 A000063 A000064

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 03 2000

It would be nice to have a more precise definition! - N. J. A. Sloane, May 26 2007

STATUS

approved

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Last modified August 18 03:13 EDT 2017. Contains 290682 sequences.