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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

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

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

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]

Peter J. Taylor, Python program to compute terms for this and related sequences

Eric Weisstein's World of Mathematics, Tangent Number

FORMULA

From Sean A. Irvine, Mar 26 2012, corrected by Peter J. Taylor, Sep 26 2017: (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...; or by L_-a(2n)= (1/2)*(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:

d(1,n)=A000182(n)

d(m^2,n)=(1/2) * m^(2n-1) * (m*prod_(p_i|m)(p_i^(-1)))^(2*n) * prod_(p_i|m)(p_i^(2*n)-1) * d(1,n)

Otherwise write a=bm^2, b squarefree, then d(a,n)=m^(2n-1) * (m*prod_(p_i|m)(p_i^(-1)))^(2*n) * prod_(p_i|m)(p_i^(2*n)-jacobi(b,p_i)) * d(b,n) with d(b,n), b squarefree determined by equating the recurrence

D(b,n)=sum(d(b,n-i)*(-b^2)^i*C(2n-1,2i),i=0..n-1)with the case-wise expression

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)

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

PROG

(Python) See Taylor link.

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 November 19 07:02 EST 2017. Contains 294915 sequences.