A000061 Generalized tangent numbers d(n,1).
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
Offset: 1
Keywords
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 689-694.
- 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
Programs
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Mathematica
Table[rowA235606[n, 1][[1]], {n, 60}] (* see A235606 *) (* Matthew House, Nov 10 2024 *)
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Python
# See Taylor link.
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
Sequence gives a(n)=d(n,1). (End)
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