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 A326603 E.g.f.: Sum_{n>=0} (x^(2*n-1) + 1)^n * x^n * exp(-x^(2*n+1)) / n!, an even function. 2

%I #6 Aug 10 2019 04:11:56

%S 1,4,-22,2,60482,-4838398,119750402,7264857602,-348713164798,

%T 4268249137152002,-2020660279013375998,327387513566969856002,

%U -17234677825923317759998,20004536762232422400002,12702000644400049997414400002,-4420880996869850977271807999998,55184528880536814259678111334400002,-98410947805609541237532965260492799998,61998887798316869577999908025139200000002

%N E.g.f.: Sum_{n>=0} (x^(2*n-1) + 1)^n * x^n * exp(-x^(2*n+1)) / n!, an even function.

%H Paul D. Hanna, <a href="/A326603/b326603.txt">Table of n, a(n) for n = 0..260</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!, where

%F (1) A(x) = Sum_{n>=0} (x^(2*n-1) + 1)^n * x^n * exp(-x^(2*n+1)) / n!,

%F (2) A(x) = Sum_{n>=0} (x^(2*n-1) - 1)^n * x^n * exp(+x^(2*n+1)) / n!.

%e E.g.f.: A(x) = 1 + 4*x^2/2! - 22*x^4/4! + 2*x^6/6! + 60482*x^8/8! - 4838398*x^10/10! + 119750402*x^12/12! + 7264857602*x^14/14! - 348713164798*x^16/16! + 4268249137152002*x^18/18! + ...

%e such that

%e A(x) = exp(-x) + (1 + x)*x*exp(-x^3) + (1 + x^3)^2*x^2*exp(-x^5)/2! + (1 + x^5)^3*x^3*exp(-x^7)/3! + (1 + x^7)^4*x^4*exp(-x^9)/4! + (1 + x^9)^5*x^5*exp(-x^11)/5! + (1 + x^11)^6*x^6*exp(-x^13)/6! + ...

%e also

%e A(x) = exp(+x) - (1 - x)*x*exp(+x^3) + (1 - x^3)^2*x^2*exp(+x^5)/2! - (1 - x^5)^3*x^3*exp(+x^7)/3! + (1 - x^7)^4*x^4*exp(+x^9)/4! - (1 - x^9)^5*x^5*exp(+x^11)/5! + (1 - x^11)^6*x^6*exp(+x^13)/6! + ...

%o (PARI) {a(n) = my(A = sum(m=0,2*n+1, (x^(2*m-1) + 1 +O(x^(2*n+2)) )^m * x^m * exp(-x^(2*m+1) +O(x^(2*n+2))) / m! ));

%o (2*n)!*polcoeff(A,2*n)}

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

%o (PARI) {a(n) = my(A = sum(m=0,2*n+1, (x^(2*m-1) - 1 +O(x^(2*n+2)) )^m * x^m * exp(+x^(2*m+1) +O(x^(2*n+2))) / m! ));

%o (2*n)!*polcoeff(A,2*n)}

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

%Y Cf. A326602.

%K sign

%O 0,2

%A _Paul D. Hanna_, Aug 08 2019

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Last modified December 5 21:07 EST 2023. Contains 367594 sequences. (Running on oeis4.)