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A005149 Sequence of coefficients arising in connection with a rapidly converging series for Pi.

%I A005149 M5168 #40 Jan 31 2022 01:32:07
%S A005149 1,-24,852,-35744,1645794,-80415216,4094489992,-214888573248,
%T A005149 11542515402255,-631467591949480,35063515239394764,
%U A005149 -1971043639046131296,111949770626330347638,-6414671157989386260432,370360217892318010055832,-21525284426246779936288192
%N A005149 Sequence of coefficients arising in connection with a rapidly converging series for Pi.
%D A005149 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005149 Georg Fischer, <a href="/A005149/b005149.txt">Table of n, a(n) for n = 1..100</a>
%H A005149 M. Newman and D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1984-0725996-9">On a sequence arising in series for pi</a>, Math. Comp., 42 (1984), 199-217.
%H A005149 <a href="/index/Ph#Pi314">Index entries for sequences related to the number Pi</a>
%H A005149 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A005149 REVERT(A014103). a(n) = -(-1)^n * A195130(n).
%e A005149 x - 24*x^2 + 852*x^3 - 35744*x^4 + 1645794*x^5 - 80415216*x^6 + 4094489992*x^7 + ...
%t A005149 CoefficientList[InverseSeries[Series[x*Product[(1+x^k)^24, {k, 1, 16}], {x, 0, 16}], x], x] // Rest (* _Jean-François Alcover_, Mar 29 2011 *)
%t A005149 a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ q Product[ 1 + q^k, {k, n}]^24, {q, 0, n}], x], {x, 0, n}]] (* _Michael Somos_, Sep 11 2011 *)
%o A005149 (PARI) {a(n) = if( n<1, 0, polcoeff( serreverse( x * prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^24), n))}
%Y A005149 Cf. A005148, A014103, A195130.
%K A005149 sign,nice
%O A005149 1,2
%A A005149 _Simon Plouffe_ and _N. J. A. Sloane_
%E A005149 Formula and more terms from _Michael Somos_, Nov 24 2001
%E A005149 Signs corrected, Dec 24 2001