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%I #18 Feb 05 2024 11:20:02
%S -1,-3,1,1,-1,5,-19,-9,41,-103,17,289,-169,331,-689,-4991,3999,7833,
%T -6509,21827,-22165,-87637,119441,-190981,-152513,1482023,-425985,
%U -1045091,1071237,-14108791,5845271,39852203,-35832801,54451699,44061359,-435442725,261309855,-22217917
%N Numerators of an infinite series that converges to the negative inverse of Backhouse's constant (A088751).
%C Whittaker's root series formula is applied to 1 + Sum_{k>=1} prime(k) x^k. The following infinite series that converges to the negative inverse of Backhouse's constant (-x) is obtained:
%C x = -1/(1*2) - 3/(2*1) + 1/(1*1) + 1/(1*2) - 1/(2*3) + 5/(3*7) - 19/(7*10) - 9/(10*13) + 41/(13*21) - 103/(21*26) + 17/(26*33) + 289/(33*53) ...
%C The denominators of the infinite series are obtained by multiplying the absolute values of 2 consecutive terms from the sequence A030018.
%H E. T. Whittaker and G. Robinson, <a href="https://archive.org/details/calculusofobserv031400mbp/page/n139/mode/2up">The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.
%F a(1) = -1.
%F For n > 1, a(n) = -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n))), where c(0)=1 and c(n) is the n-th prime number.
%e a(1) = -1;
%e a(2) = -3;
%e a(3) = -det ToeplitzMatrix((3,2),(3,5)) = 1;
%e a(4) = -det ToeplitzMatrix((3,2,1),(3,5,7)) = 1;
%e a(5) = -det ToeplitzMatrix((3,2,1,0),(3,5,7,11)) = -1;
%e a(6) = -det ToeplitzMatrix((3,2,1,0,0),(3,5,7,11,13)) = 5;
%e a(7) = -det ToeplitzMatrix((3,2,1,0,0,0),(3,5,7,11,13,17)) = -19.
%Y Cf. A088751, A030018, A072508.
%K sign
%O 1,2
%A _Raul Prisacariu_, Jan 08 2024
%E a(21)-a(38) from _Stefano Spezia_, Jan 09 2024
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