Search: id:A006232
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%I A006232 M5067
%S A006232 1,1,1,1,19,9,863,1375,33953,57281,3250433,1891755,13695779093,
%T A006232 24466579093,132282840127,240208245823,111956703448001,
%U A006232 4573423873125,30342376302478019,56310194579604163
%V A006232 1,1,-1,1,-19,9,-863,1375,-33953,57281,-3250433,1891755,-13695779093,
%W A006232 24466579093,-132282840127,240208245823,-111956703448001,
%X A006232 4573423873125,-30342376302478019,56310194579604163
%N A006232 Numerators of Cauchy numbers of first type.
%C A006232 -a(n+1), n>=0, also numerators from e.g.f. 1/x-1/ln(1+x), with denominators 
               A075178(n). |a(n+1)|, n>=0, numerators from e.g.f. 1/x+1/ln(1-x) 
               with denominators A075178(n). For formula of unsigned a(n) see A075178.
%C A006232 The signed rationals a(n)/A006233(n) provide the a-sequence for the Stirling2 
               Sheffer matrix A048993. See the W. Lang link concerning Sheffer a- 
               and z-sequences.
%C A006232 Cauchy numbers of the first type are also called Bernoulli numbers of 
               the second kind.
%D A006232 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006232 A. Adelberg, 2-adic congruences of Norland numbers and of Bernoulli numbers 
               of the second kind, J. Number Theory, 73 (1998), 47-58.
%D A006232 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
%D A006232 H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, Cambridge, 
               1946, p. 259.
%D A006232 Ming Wu and Hao Pan, Sums of products of Bernoulli numbers of the second 
               kind, Fib. Quart., 45 (2007), 146-150.
%H A006232 T. D. Noe, <a href="b006232.txt">Table of n, a(n) for n=0..100</a>
%H A006232 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A006232.text">
               Sheffer a- and z-sequences. </a>
%H A006232 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BernoulliNumberoftheSecondKind.html">Link to a section of The World 
               of Mathematics.</a>
%F A006232 Numerator of integral of x(x-1)...(x-n+1) from 0 to 1.
%F A006232 E.g.f.: x/log(1+x).
%F A006232 Numerator of Sum_{k=0..n} A048994(n,k)/(k+1). [From Peter Luschny (peter(AT)luschny.de), 
               Apr 28 2009]
%e A006232 1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90,...
%p A006232 seq(numer(add(stirling1(n,k)/(k+1),k=0..n)),n=0..20); [From Peter Luschny 
               (peter(AT)luschny.de), Apr 28 2009]
%Y A006232 Cf. A006233, A002206, A002207, A002208, A002209, A002657, A002790.
%Y A006232 Sequence in context: A040345 A070685 A033339 this_sequence A122549 A039942 
               A050276
%Y A006232 Adjacent sequences: A006229 A006230 A006231 this_sequence A006233 A006234 
               A006235
%K A006232 sign,frac,nice
%O A006232 0,5
%A A006232 N. J. A. Sloane (njas(AT)research.att.com).

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