A326556 - OEIS

%I #6 Jul 28 2019 00:27:27

%S 1,-4,256,-67072,49479680,-82817122304,273099601739776,

%T -1606512897507196928,15659025634284911198208,

%U -238894370882781809622384640,5451274531297360096585324691456,-179296966081016547805899589056200704,8242844472527700570663352676068232265728,-516102091343047279882754030489835708929277952,43042816831864259208854418353099287467922680709120

%N E.g.f. C(x)^2 = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!^2, where C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ) is the e.g.f of A326551.

%C The e.g.f. C(x)^2 can be derived from the functions described by A326800, A326801, and A326802.

%e E.g.f.: C(x)^2 = 1 - 4*x^2/2!^2 + 256*x^4/4!^2 - 67072*x^6/6!^2 + 49479680*x^8/8!^2 - 82817122304*x^10/10!^2 + 273099601739776*x^12/12!^2 - 1606512897507196928*x^14/14!^2 + 15659025634284911198208*x^16/16!^2 - 238894370882781809622384640*x^18/18!^2 + 5451274531297360096585324691456*x^20/20!^2 + ...

%e where C(x) is the e.g.f. of A326551:

%e C(x) = 1 - 2*x^2/2!^2 + 56*x^4/4!^2 - 8336*x^6/6!^2 + 3985792*x^8/8!^2 - 4679517952*x^10/10!^2 + 11427218287616*x^12/12!^2 - 51793067942397952*x^14/14!^2 + 400951893341645930496*x^16/16!^2 - 4975999084909976839454720*x^18/18!^2 + 94178912073481319162642169856*x^20/20!^2 -+ ...

%e such that C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ),

%e note also C(x*y) = cos( Integral Integral C(x*y) dx dy ).

%o (PARI)

%o {a(n) = my(C=1, S=x); for(i=1, 2*n,

%o S = intformal( C/x * intformal( C +x*O(x^(2*n)) ) );

%o C = 1 - intformal( S/x * intformal( C +x*O(x^(2*n)) ) ); ); (2*n)!^2*polcoeff(C^2, 2*n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A326551,  A326800, A326801, A326802.

%K sign

%O 0,2

%A _Paul D. Hanna_, Jul 28 2019