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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 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 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

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Last modified September 20 16:31 EDT 2020. Contains 337265 sequences. (Running on oeis4.)