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%I #28 Mar 06 2018 11:38:46
%S 1,372,29250,-134120,54261375,-6139293372,854279148734,
%T -128813964933000,20657907916144515,-3469030105750871000,
%U 603760629237519966018,-108124880417607682194048,19820541224206810447813500
%N Convolution square root of A000521.
%C Triangle A161362 = the corresponding convolution triangle with row sums = A000521.
%H Seiichi Manyama, <a href="/A161361/b161361.txt">Table of n, a(n) for n = 0..426</a>
%F Given A000521: (j = 1/q + 744 + 196884q + 21493760q^2 + 864299970q^3 + ...); multiply by q and take the convolution square root.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = f(t) where q = exp(2 Pi i t). - _Michael Somos_, May 03 2014
%F G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/2). - _Seiichi Manyama_, Jul 02 2017
%F a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(5/2), where c = 0.378271951998085144930610869223050101960774818... = 3^(5/2) * Gamma(1/3)^9 / (2^(7/2) * exp(sqrt(3) * Pi/2) * Pi^(13/2)). - _Vaclav Kotesovec_, Jul 03 2017, updated Mar 06 2018
%F a(n) * A299832(n) ~ 3*exp(2*sqrt(3)*Pi*n) / (2*Pi*n^2). - _Vaclav Kotesovec_, Feb 20 2018
%e a(2) = 29250 = 1/2 * (A000521(2) - 372^2) = 1/2 * (196884 - 138384) = 29250.
%e G.f. = 1 + 372*x + 29250*x^2 - 134120*x^3 + 54261375*x^4 - ...
%e G.f. = 1/q + 372*q + 29250*q^3 - 134120*q^5 + 54261375*q^7 + ...
%t CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(3/2) / (4096 * QPochhammer[-1, x]^12), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 23 2017 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = x * (eta(x^2 + A) / eta(x + A))^24; polcoeff( sqrt(x * (1 + 256*A)^3 / A), n))}; /* _Michael Somos_, May 03 2014 */
%Y Cf. A000521, A161362, A192731.
%Y (q*j(q))^(k/24): A289397 (k=-1), A106205 (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304(k=10), A289305 (k=11), this sequence (k=12).
%K sign
%O 0,2
%A _Gary W. Adamson_, Jun 07 2009
%E More terms from _R. J. Mathar_, Jun 15 2009
%E Keyword:sign introduced by _R. J. Mathar_, Jul 07 2009
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