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%I #19 Nov 14 2023 11:39:49
%S 1,6,2268,7434504,95227613712,3354162536029536,264444869673131894208,
%T 40740588107524550752746624,11136881432872615930509713801472,
%U 5026062205760019668688216299061782016
%N Coefficient of x^(4n)/(4n)! in the Maclaurin expansion of cm4(x), which is a generalization of the Dixon elliptic function cm(x,0) defined by A104134.
%C Equals column 0 of triangle A357801.
%H Vaclav Kotesovec, <a href="/A153300/b153300.txt">Table of n, a(n) for n = 0..117</a>
%H Alessandro Gambini, Giorgio Nicoletti, and Daniele Ritelli, <a href="https://doi.org/10.1007/s10013-023-00617-3">The Wallis Products for Fermat Curves</a>, Vietnam J. Math. (2023).
%H Alessandro Gambini, Giorgio Nicoletti, and Daniele Ritelli, <a href="https://doi.org/10.1007/s00025-023-02038-7">A Structural Approach to Gudermannian Functions</a>, Results in Mathematics (2024) Vol. 79, Art. No. 10.
%F Define sm4(x)^4 = cm4(x)^4 - 1, where sm4(x) is the g.f. of A153301, then:
%F d/dx cm4(x) = sm4(x)^3 ;
%F d/dx sm4(x) = cm4(x)^3 .
%F a(n) ~ 2^(14*n + 11/4) * Gamma(3/4)^(8*n+1) * n^(4*n) / (exp(4*n) * Pi^(6*n + 3/4)). - _Vaclav Kotesovec_, Oct 06 2019
%e G.f.: cm4(x) = 1 + 6*x^4/4! + 2268*x^8/8! + 7434504*x^12/12! + 95227613712*x^16/16! +...
%e sm4(x) = x + 18*x^5/5! + 14364*x^9/9! + 70203672*x^13/13! + 1192064637456*x^17/17! +...
%e These functions satisfy: cm4(x)^4 - sm4(x)^4 = 1 where:
%e cm4(x)^4 = 1 + 24*x^4/4! + 24192*x^8/8! + 140507136*x^12/12! + 2716743794688*x^16/16 +...
%e RELATED EXPANSIONS:
%e cm4(x)^2 = 1 + 12*x^4/4! + 7056*x^8/8! + 28340928*x^12/12! + 419025809664*x^16/16! +...
%e sm4(x)^2 = 2*x^2/2! + 216*x^6/6! + 368928*x^10/10! + 3000945024*x^14/14! +...
%e cm4(x)^3 = 1 + 18*x^4/4! + 14364*x^8/8! + 70203672*x^12/12! + 1192064637456*x^16/16! +...
%e sm4(x)^3 = 6*x^3/3! + 2268*x^7/7! + 7434504*x^11/11! + 95227613712*x^15/15! +...
%e DERIVATIVES:
%e d/dx cm4(x) = sm4(x)^3 ;
%e d^2/dx^2 cm4(x) = 3*cm4(x)^3*sm4(x)^2 ;
%e d^3/dx^3 cm4(x) = 6*cm4(x)^6*sm4(x) + 9*cm4(x)^2*sm4(x)^5 ;
%e d^4/dx^4 cm4(x) = 6*cm4(x)^9 + 81*cm4(x)^5*sm4(x)^4 + 18*cm4(x)*sm4(x)^8 ;...
%t With[{n = 9}, CoefficientList[Series[JacobiDN[Sqrt[2] x^(1/4), 1/2]/Sqrt[JacobiCN[Sqrt[2] x^(1/4), 1/2]], {x, 0, n}], x] Table[(4 k)!, {k, 0, n}]] (* _Jan Mangaldan_, Jan 04 2017 *)
%o (PARI) {a(n)=local(A);if(n<0,0,A=x*O(x);for(i=0,n,A=1+intformal(intformal(A^3)^3));n=4*n;n!*polcoeff(A,n))}
%Y Cf. A104134; A153301, A153302 (cm4(x)^2 + sm4(x)^2).
%Y Cf. A153303 (cm4(x)^4), A357801.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 02 2009
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