Search: id:A002206 Results 1-1 of 1 results found. %I A002206 M5066 N2194 %S A002206 1,1,1,1,19,3,863,275,33953,8183,3250433,4671,13695779093, %T A002206 2224234463,132282840127,2639651053,111956703448001,50188465, %U A002206 2334028946344463,301124035185049,12365722323469980029 %V A002206 1,1,-1,1,-19,3,-863,275,-33953,8183,-3250433,4671,-13695779093, %W A002206 2224234463,-132282840127,2639651053,-111956703448001,50188465, %X A002206 -2334028946344463,301124035185049,-12365722323469980029 %N A002206 Numerators of logarithmic numbers (also of Gregory coefficients G(n)). %C A002206 For n>0 G(n)=(-1)^(n+1)*int(1/[(ln^2(x)+Pi^2)*(x+1)^n],x=0..infinity). G(1)=1/2 for n>1 G(n)=(-1)^(n+1)/(n+1)-sum((-1)^k*G(n-k)/(k+1),k=1..n-1). Euler constant is given by : gamma=sum((-1)^(n+1)*G(n)/n,n=1..infinity) [From Groux Roland (roland.groux(AT)orange.fr), Jan 14 2009] %D A002206 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A002206 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A002206 E. Isaacson and H. Bishop, Analysis of Numerical Methods, ISBN 0 471 42865 5, 1966, John Wiley and Sons, pp. 318-319 - Rudi Huysmans (rudi_huysmans(AT)hotmail.com), Apr 10 2000 %D A002206 Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 266. %D A002206 A. N. Lowan and H. Salzer, Table of coefficients in numerical integration formulae, J. Math. Phys. Mass. Inst. Tech. 22 (1943), 49-50. %D A002206 H. E. Salzer, Table of coefficients for repeated integration with differences, Phil. Mag., 38 (1947), 331-336. %D A002206 P. C. Stamper, Table of Gregory coefficients, Math. Comp., 20 (1966), 465. %D A002206 Ming Wu and Hao Pan, Sums of products of Bernoulli numbers of the second kind, Fib. Quart., 45 (2007), 146-150. %H A002206 T. D. Noe, Table of n, a(n) for n=-1..100 %H A002206 G. M. Phillips, Gregory's method for numerical integration, Amer. Math. Monthly, 79 (1972), 270-274. %H A002206 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A002206 Index entries for sequences related to logarithmic numbers %F A002206 G.f.: 1/log(1+x). %F A002206 G(0)=0, G(n)=Sum_{i=1..n} (-1)^(i+1)*G(n-i)/(i+1)+(-1)^(n+1)*n/((2*(n+1)*(n+2)). %F A002206 a(n)=A002206(n)/A002207=1/n! sum_{j=1}^{n+1} bernoulli(j)/j S_1(n, j-1), where S_1(n, k) is the Stirling number of the first kind. - Barbara Margolius (b.margolius(AT)csuohio.edu), 1/21/02 %e A002206 Logarithmic numbers are 1, 1/2, -1/12, 1/24, -19/720, 3/160, -863/60480, 275/24192, -33953/3628800, 8183/1036800, -3250433/479001600, 4671/ 788480, -13695779093/2615348736000, 2224234463/475517952000, ... = A002206/A002207 %e A002206 G(0), G(1), ... = 0, 1/2, -1/12, 1/24, -19/720, 3/160, -863/60480, 275/ 24192, -33953/3628800, 8183/1036800, -3250433/479001600, 4671/788480, -13695779093/2615348736000, 2224234463/475517952000, ... = A002206/ A002207 %p A002206 series(1/log(1+x),x,25); %p A002206 with(combinat,stirling1):seq(numer(1/i!*sum(bernoulli(j)/(j)*stirling1(i, j-1),j=1..i+1)),i=1..24); %Y A002206 Cf. A002207, A006232, A006233, A002208, A002209, A002657, A002790. %Y A002206 Sequence in context: A040353 A128160 A092120 this_sequence A040349 A040350 A089572 %Y A002206 Adjacent sequences: A002203 A002204 A002205 this_sequence A002207 A002208 A002209 %K A002206 sign,frac,nice %O A002206 -1,5 %A A002206 N. J. A. Sloane (njas(AT)research.att.com). Search completed in 0.002 seconds